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dt{Jl g 5 -rr-l-fu 4.-.J^t st ;-Fb,5l,r&U -$ dlilr bV 9f t o)!..J11*
.!iU r;-1, J \;3 C,,|,..q:.:, g;bl); r5Lr&ti
.€e.Jbl L J31;J 6rlf*'r1jJ-,itd jl #
r!U6l^* ca,2,.t-rrb/ = g@) ; asx<boKy' .(.-Q- dt#-J d) \ \A.63i
limI-o
I0)
I
b=l
b
g(x) sin k dx = "
g(x) sin k dx
gie.sLJl(\ )
4D= I sin )'x dx
*+) G)
s-fr-t" ^s(r*1to
T c'\-- I
,*t-r it JlFl .1rr + + q r,!r.t tl
(Y)
f;l: (Y) -r (\ ) d'trt g.+ t:
b u-{rI(I) = ["- s@ sin )x dx { *r s&
1
sin k dx
qrts)!-rjrt{xjs$(_k- . (rf )
eJd cr: ? b.s-tLi d-h JV
rr(1)=- I" g@* | lri,,t"* *1u ggrrrr** . f '"Is@-s.,+
+ )J siilxdx
"'+ ^ t'f a
r-rr Lgljl ! e-.r-y e.rl;+ ;; tlt;\Sg(x) ,Gil € ,?f u;4,r^,lt lt7@) {#tt"+a# ;olg6)l s U ,',;lr M tc,:Vr:,tr
bt!u.f
I {" + s&+ f )rint"* I = I ["
* f s@)sinJ.xdx I
.oM),
J-J+-a J
s(x) siilxdx t =41
till jl
l s@ 's(x +
4+^:i J)
l4D I
I Clr5'; u34gl-r Slt6y #jlJ-b q;; .L"tcfcrt,ilq qi,ai ,rlc._f {d.,al'i .ta- 4;Vl,..oU:l 4, -f cgll b *ii c.-,1 ,Sg .rh Al Sll.r:,-,1 j-tt
| \^) t =ry. + {' *, s&) . s@ * # ) | dx
lF- t* q{rti alas@) cr-t rtf .:-,1 ,,:-',3l-[a,b] E,."1d.rU $(x) 13-l u. rbt
. f-Jl Jli g st* lrl. ,'U,-6)Jre c.-L )Jr y' ,sl;l q i+i r) .)j JrlF .:.r'r4
xrt.s5l q,ryq s , XlX.-t,sljlq.tr-s
ftt.#+9=,
YY
(r0) JJlJ".i
.C=5rlF ,-:,-.,1 tu3-t Jilg 6 [a,b]
Yrr ./.rb !.lrrl* ?Jfs 5.215-t.5h* [-n , n] ta{Lrt! tf@'6y' 'Y 4';ix r, gl;l q .t*i r-.lb
! + i (an cos ru * bn sin ru) = ] fA*) + f(x-) lY n=r
.r, Cb fr tr:-,,|, );J- -tl-t 4t +f@-) sf@+) s -[sl *-lt'bn tan ..li -;: 6..f::....OX af-ti.j
€eroGy'.gLJl
s* (x)A. .
=- ?Y
k
7,ro" cos nx * bn sin w)
\,t I "fAl cos n t dt , * !" ral sin nt dt
d)-Scr-Fu q.6"i
p*t-.f bn tan 13t? q 4i +
tf !"fftlcosnt dtlcosruq Ihrl sin nt dtl sin rtx}-Jt
s*(x)=+ i fato,**j,
+ *n'l+k
+ 2 cosnt cosra,8= I
s*(x)= * sin nt sin wl dt
r,_.r,rt]t,.r"'.rtdtFl ,V_r . 1tf )
s*(x)= + I"ft,l/ + *,f, cosn (t - x) I dt
$in gcos na = sin (n +*r, - sin tn - | I, grbqqitl
t'*t'rwingr+.:
:^nol = sinL + [ sin+ sk+ l+... + [ sin (k ++;
-sinG + )al =sin(k*|t"
t+i;lr
sin (k + + ) ft - x)dt
filrJ = t -x,iaJ=t'ti{'t lnn' ' lsin( I'.! 1
. z-x sin(k+It,s*(x)=; Ift'+x) +Ysrn 1
I,1
.l'<ii -r-iL Yn ./rLs otlr ! ,-:9Vf(x).6y'
srnfk+*)s dss*(x) = * ['f6 + x)
sin(k+])t^+t ? ..:-,,1
I f yr,lK,-y-:,j f ++e d(,-y : J -r t + J -r t u-lYsm -Y
s*(x): + I"ttrl
ds
Ysrn IY
* !,"
1,*t-f f-^) .rli -r-l.l itj..f .Jt? q
ds = L I" t+ .j, cosn sJds=\ (\ )
g-t,l tffi l.rsr(x) Ju
s*(x)=+ !1r*r1 "' c+ * r' ds +* Ift *n 'i t
* *' 'r,tu\/ n'-;\ r ysrnl fr-o' ysrn-
YY
sin(k+*)t^['=I ['111x+s)-f(x-)+f(x-)J =a'" ds-f fi ln'n.
rr ' ttt " vsrnl
Y
C*t-.r (\ ) q .?i t:
sin(k*|ttrsur I
Y
L I'fa-)fr-n
r. =tlD- + \'Yrt
i-Jl
sin (k * {t tds-k)-r.I .fY v)n
ds = f(x')sin(k*{t'
Ysllr Y
cr-trtr
vsrn IY
*|t' ds
J
Y
o
I-tt
f(x+s) - f(x-) sin (kvsln I
Y
6,, f(x+s)-f!x-) = fim,f@+s)-f(x-)/ + um If@+s)'f(x') t =f d-i;;:- ysrn I r{.- s ysrn ..{- r*.-' r
YY
5 1,31- r^,, d I r ^",u s r-l )1..:-, 1.,1 I -.-r al &s t * rWC b cr-lr trYsm -Y
rt; q+ t tJ.",t5q Jli,l
limk+o
f(x+s)-f(x-) sin(k+I )sds= o
Ysln I ' Y
Y
q.ry st.r.., r kJrFr ,4-r ' ( lA )
1+::t J)
= + r@+)-a'*3Y n=l
(an cosra * bn sin ru)
yhae
.''rlJt{
+ f(x-)l
J|-r.i, .li q o+y ! -l *it--rr sx<rt d^eU .trf6) = x * tt Cb t's! 6f 'V Jtb.r.tl* b \ + +. i. ... cerd)r'
.Sr-5crli.lly.rr FL t :,Fx + xt - {* f ( J,. (- \),, cosra- } C ln sin w)
f r=r \ n\ n
i@) = tt * aY r,3-l tto.i t .eot-"r x=lt t, l.r Cb t':! 6r.rl& iJFl (r'
g:ltf(n+)='n+nl
, * $ t1)n cosr,.= + tf@+)+f(n-)l= + [@ +rt')+('n +,t')] =,t'
..:-,1 f.-^, ,IU ; Yl ,-,11s or.ll tt Cb r" AU J'V r' +5 ddl
i,*=''*+.+*"'=!
'y*I'=rylim r' : f(x+)k+o Y
lim. s*(x)&'+o
L+13 JJ
)L.i l5 €e +j t g#,lf@) A1'x ,,Li';.rt l.r Cb 4,.*rt tSr rt q crrljl
..:,-,1- :L., .JL^ ,p +t t-lj cSf -2.: gilLj
:f(-n) = f(n) J o),a [-n , n] i.,,lt.;) r:-r.;r e.Uf(x) +tS utt .f 4..,ii
! .rrir \f @) Cb.rt csr.Kji .$u 13;h* $a'l3-t {ru ,t-l if @ ;r;a,.'a
drr-f jl JirL- tS* S .tUT r.a-r.l y,f(x) u-t! cSf ;l rf.; r, 4l^? 6$';t-g-.-,, trx thi;y' ..1l 6,;i*-
..>tt'-^* rt Fil + r&-)I
a.Y
,64.. \l-+LY n=l
A.=+ $@)dx=* tf@)-f(-n)t =.
An= \,t
f';lr
I7 g cosra f,a - | ffO cosrulltn+
-tt lt I fO sinru dx-rg
.r-iL J: ott'l t-.rrll e|-rdbl) 4j ^,.f :f €,; o.l .gLJl
+ > (an cosru * bn sin nx)tt= t
(An cosnx * Bn sin ru)
r,_r-rr i.,)t1rr r t+Jtft;t ,4.r . ... ( Y. )
An=nbn; n= \-lY-ll'-l ...
t^ = * {{O, sin ru tv - } fO sin ruJ ; * [7o cos ru dx
Bn=-flan; n= \-lYr...
t ll.- or.eli jp c.-; *xf &) 6 *E rr .t_id Jirt- cs;t u-pi Gr ijtu- \t-,'l il t 6 ;J*.1-+
,:, (n a,,sin ru .r nb,,cosru) = + lf @+) + f (x-)]
.,,1 .Kl;i.r,.iL Yrr otlr !.:-lL:.r dta53 tc-,t;^t[-n , nl tf@ #,-it.f 4-,ai
;t;1f&) +,sl 6r
t's1o ,Sf .t-lr *-t+/ (x) 6 ;blit.rr i'-lftf
f (x) = + *,,:, (An cos ta + Bn sin rc)
+i C i J; Jsj,e t1
@
f (x) -
u;! 6f;l rl-, q d-+ 6S;*U. S
f(x) - ? .,:, (an cos ra * bn sin ru)
(vt 1 ... Jlt J.ai
@
A + E (a,, cosnx * b,, sin ru)Y r=l
.tf ,s-$,t,S:t d-rti f i {t? {, d.? JliC
"J*+U -r,-! 6rlj;r .gtJlI - -'\-'
b 'o+**f IIo(ancosru*bnsinra)dvlI rgf,a- { Y ,,=, a
f^^) ,-lL:; c-,,1 .,jtS ,l-l.''\ s
f ro, i I dx -,t + [ @n sin nb - sin na) - b,, (cosnb - cos na)l
r+r,,/ )l; * c--i.j
F(x)= { tfq-lto,
,i f o;)tc S:t$l.1,- ^:''t-,F.:*,1 .SlnSs *:*,f r,(.!-l +. tb;
F'(x)=f(4-?
,. = ] I" f@) dx llnotlr ! ,=sdfr(!-l .,;+j! Jr-.c*,1 dl..* uf,lt -n
f--rl'r
F(x+yn)= ( rt,t-\ rc, * t'iiit I to'-F(x). Iir,t-lto, - F(x)
rr-1y .i$-.r.j, tdtFt.t1,_r . ( YY )
.-r-:!f call* catS.j:5 cSbl; r5.:*,1 ln i22stf .f.lt - 9 u7-t ,-;1.VF d/-lJt l
F(x) (An cos nx * 8,, sin ru)
F(x)
b
! rvdt={a
f(,) d, - { f0 dt
.6
=L+IY tt=r
=+ *:, t + (a,, sin ra'
i rat dt - ?'** *
i ,o,*= ? @-a)*":,
i ,o,*=?@'a)*,:, *,o^
cr-rr:u: rBn = ff /, = - *,.ri.,r e5
bn cos ru) l
6 ..J C qr3 F(x) -i,*f ., tf
; L @,, sin nx - bn cos ra),:t n
{<r-l \ -b
t-! 1a,sin nb-b,rcos nU-2n ^=r
f;b
L gsin na-bncos na)n
o)i;Crir." Jligtg^J- O{l iJ.L, ,t'-lfd.:lp e,
(sin nb - sin na) - bn @os nb - cos na)J
..:,.,,1 6;J|Stl d-? q d.? \f(x) u:! cgrjl J.rb cgr 6
(lr) ""JrlJ"i
!l.r*r* \ f(x) = siru; o1x1trt Cb q Pv ,-t-*s +';! tsr y\ JtI
.+k lscos, db qu;.r tSr di ;l 6rKl;l'bn= o
1l-
f(x)cosru d* = L
flsinxcog&= o
[n ,, x cosnx fl;6 -- v/ t + (- ])'/ ; n* \* n( \-n')an=L I
IE
c, = Jl- I,t
sinx = -lL + jl- t,t ,t
OAsrcosx =-- afi n-,
cos lrrJ\ - frl-
n sin lru\-frl- '
c 1X1trt
filr .:-,1 P tt 6,Wt c;ljl q an tS:'-l n -b
; " 1x1rt
1'*t-f ,S ! c*!tr
JeV t7r :l S-3rt.$l tr-r -r.rt* l.2f$) = x ;
x = y I sinx - ry * tin:' - ... IYT
| [-cou.ry # +...]+c
i* =, [o"*
-Jt<x<lt t;! tSf .1 JU,
.J.tk trtt Cb +,-s! tSf
r*tr4s;Jtfst\x'Y
fit-fi
".l-.l t gle JlFl n J-n d-rU.;: giilrj U-l ,yI-h ;lc ;tzl- 6lta
'lr?
q.r-rr .^i)!_.r+.r t+JtFt ,4_r ... 1 vf )
t+,i;l-r
d/-lJq
.r'= $. t,i + cosra
C6.lt t-.l39 cSf .Y.\
tp_,-t*iLr Yrr .7rb.tlr Q J.7jL;. *L.lr ! c-i f(x,y) J+-f rir-;;11.q.g"i l-iLrl,.-r jt5o;l"ul t Clr s f@+yn,y) = f(x,y + fn) = f(x,y)
C*t-.f r 1c.*;f@/) gb +'s! ,Sf
:,*, [tt,n 0) cosr/N * b,n g) sinmrJ
..li;r 5
aur9) I f@,y) cosmx dx
h,,(!) = * !" f4,tl sin mx rtx
1.,, tgi J rr-^ y.lt - bu,$) sar,(f) d/-lJt+ .:-.,1 =st;-y
u '>*;f@y) &:,-l \ rbJ
rl.r F_L.r q's! tSf otf + JliC i
(oru, cos ny * b,,r, sin ny)
flx,y) _ a.(y) *Y
-\xt
@
-orn., SL
Y ,:rarr(!)
(Yo) ""Jrl J''i
b^(y) (c^n cos ny + d,*, sin ny)@
-C^., lFL
Y n=l
,.li -2.t 6
( talo,,* = * !*"
b,*,= * !," I\cmn-1
JT'
d^r=)Jt'
f(x,y):?*+,:,
€
z=l
!fWcosmx cosnydxdy
" flty) cos mx sin ny dx dy
I* I 'fltr) sin m x cos ny dt dy
( tu1
( tcl
(Y)
tt tt
! I ftxyl sin mx-fl -Jl
sin ny dt dy
[a.ncos ny+b.usin "y\* i
( \d)
al;tr
3 |o,n.ro, m x*c,n;in rruvl
€+X
m=l[a,*rcos mx cos ny + b*ocos mx sin ny
+ crrrrsin mx cosny + dr*rsin mx sin nyJ
.i".-lf(x,f) 6U.u61.: ut3r ,gf 6
o-.,J, o)L-rt'.r t+JtFt ,4.r . . ( Yt )
,itea,*, -t \--l_e.-o.t*i/(x, -il = f6,il :f?qfl = f&y).+lr?
d/-lJtl
f(x,y)= ?
arr =),t'
f(x,y) -
?-,o*tot mr cos p|
(t)
o6y'
f*rlr.t*i
(0)
..li 1r 15
(v)
.li;.: rS
(^)
= -f(x,y) +tr" .rL-!.,,r
. + 2,o*'o'nY++i, am'osmx*:,
1g 1t
I I ft+yl cos mx cos ny & dy
f(x,-y) = -f&y), f(-x,y) = f(x,y)
@@
b.,rsin ny + 2ln=l ,l= I
f(+y) =\Y
@@@
tn=l tn=l tt=l
cos nu sin ny dx dy
f"*rlr ,G;i f(r,-y) = f(+y) , f(xil = -f(x,y)
_€\sY tt=r
brr* = * [' !"f$,y) (?)
fr
furut - a7t' I"n I."fltr) sin mx cos ny fu dy
f;tr .t<ji f(-xg = -f(x,y) , f(x,-y)
(Yv) ...."..r'"'' JJI J''l
(q)
.li;r +5
f(x,y)=
(\ . )
= t/ Cb .\ JU,
.J-rr V.p -rt11rt
+(!-l u. th;
f(x,-y) - -ry = -f(x,y), f(-x,y) = -ry = -f(x,y)
f-.rlr
1", y sin mxsin ny dx dY = (-\)'"*" **
66
f(x,y)= ttn=I il=!
;l c..-,1 &;t* .b ,t-f +'s-f 6f d/-lJt+
(-\)^*n ' sin mx sin nY
mn
€
m=l
€
ll= I
d-rti 2r ctS;r u-t-t' 6f Js:a + l:f&Y)
drr. = )fi'
t'fi'<x<rt
ar*r=* !
u-tPJltsJl .r.\oif f, Cb -f I rr-r;9 6f gl.rlr .:rL:' Cb r^ J^! cJ'rLjr \'\ u:''".rr ait+
.,,lt - rf Ctt eic,.-,| o#-+..:,,-,1 Ple +'t-tt eV-f t5Ll) ,r.rb,' .-.,A,r*;
y = f(x) ,' -6(x(o
,,.q1;rdr CfJ'q us3, tgr b oy..e +Jliy,VU J:c.-i t':! 6r,Jbl)
!_.r.t, s)b_.r* r t<JtFl ,4-r . ( yA )
.F-L.J JfFl $.>,p' lolj'll, tglu-2g tSfdr:i" J> J.,_r f+ff {+t ?1.:l:
CttJj U-l ,Sll..:-,,1 lyy +*t-f JlFl r,r5rlr
fi(x)=f(x) ; -l<r<l ; p=\lI
^S ;itsft@ eV -t- r5.:*,lrrg-{ t}e -b ,tl.; c^.,,1 olgJ.r:re 5,_/.1i.;r 15
.:-,,1 us-f 6f L1lj:f(x) gb .c,.."ly : f(x) ; -@<.r< * Cb JSJii, c--t+.r q
j)..J-c,-,1 tlll*o 6 ,-?t q':g 6f d/l J- \f(x) I
=timt?+ 1o,,ro, S1 * bn sin ( x))l.+o Y
('I
O
n=(x) ,
" ijl
.t<ji rr-,1 c..:ti {.trrr M Ji.rr r5.[* lf@)l dx - M < * ,-p-.r-!b
ti^ to.t=y:| ttlatdxt=t:::+ {,'tro,td* =t:,i 1irn,w*--tnU
=.
rf:r Ltt, f(x) € *b qJq .z*,tf1g) Cb.r, tSr jt u*st 6r
f-rb,:*lrt+
6
f(x) - tim {l*o tt= t
(ancos +, * bn sin ff fll
it iy drb.l.t Jlrf.sl otf t;11b,, 5an y-tlL 5wn = T f+r"/ rlf Jb
, ,l.'\CJ f=ot.r
* sinw,,x I ffq sinwntdtl\-t
@
n=If(x) -,,y+t
I
[cosw,,x I fO cosw,,tdt-t
p+r rli ^+La Jt-.
Aw = wn+t-wn
f(x)
,tI
c*t-r Aw x + ,t;.rr i;b ?-t'\ .6"i
,"i, u^ n* [,ftt)
cosw,,rdt * sinw,p {,ial
sinw,,t dt] Lw}=Lu^fr l**
=*i
a(w) f(x)coswxdx
b(w) f(x) sinwxdx
f(x) [a(w)cos w x * h(w)sinw x ] d w
lxl<o
lxl>n
(\a)
t
( \b)
p*t-rf .t*i sfrt' .[sl v-l.r- o rj
=*i
-\ i'a -oo
f;
-I
€,it Jt ;u
(Y)
..:-,1 ly:,f(x) Cb.;t JlF"l r, rf
t{ b iri 6c..,1 JlFl Lfr 4 ii Lfr t $tx twn )ld-tt *tf ,Fl: .i1Lr
f(x)€
[coswxIfOcoswtdt-o
* sr,n w x ! fO sin wt dtJ dw
f(x) =Cb.-.rt JtFt .\ Jtt
er_;y n)L_.rt' r t+JtFt ,V_r . ( r' )
.J.tt*t )-tt7 Oye 912
1-;b: J'>
@11
I lf@ldx*tI xdx=n'-@o
- \ itelcoswxdx= l- !"7t _;- Jt
b(w)= * IVel sinw ' d'= +
{p*rlr j.hll e.-:.-,1 t:+y )-r1,1Cb.;.;r JlFl d/-lJtr
a(w) xcoswxdx,= "
lt["xsinw x dx= ] t- L ror*t * * sinw xl .oo'
b(w) sin wn - n cos wn)
d/_lJtrr
f(x) - l',[-I f -L sin wn' n cos wn) sinw x d w
ltww
rb tSb .1o* ci,! ^,sg ,SV-r q b*tr:+ .,,, d .t;.iLr 6 rl ,:c J.,-t-l
b-ny d-lF-$ # r.t-tay' t) f tf ,; orLi:-,| - rL.i il +:t JlFt;: gi;tj.r.sok ,* ttliclt
^,-s! JtFt q
oK3i -r-.:b LSlaSi j3,v. t .'4l-3;;t Llt,y = f(x) ,' -€ <r< - 06y' .\ 4.;i4tA Orl-.,, Cb c.-,1, J \-7 )1J> ,i.*t" g...-, trx thit; tf$) Cb .;t JlFl
jtp- \),-"|ts "
I- to(w) cos wx * b(w) sinwx) dw = |f f@+) + f(x-)l
u-:gJlFrl s cgr -hLi, esg* .f .\
*Jq+ Jli,-lrxiq - cos |x.i sin 0 sV-rt;l ortn:-,'l U
cos nr = I Gtu + dtu), skro = * @tu - r'bo)YYT
./-lL:. eb.--rt 6f.2r y-:Ur ;r-l .1r.,5;;(1t+ tr -l
y=f(x); -nt<x<tt ; p=ltt
i,f-
f(x) = at- a 2, ,o, cosra * bn sinru)
{.*t-"r
f(x) = a. a i,, i ro,' ihn) ebo + i ro^ + ib)'-iu 1
- a. f+)"/ ',lli drii? Jb
L.--Y
dn=iro,+ib) , cn=+ @n-ibn)
f_-rl: cxt rtf
€f(x) - c. +
ll=l
(\ )
er_;y o)b-.rj r tdt,Sst ,4.r ( rY )
,rr=* (an-ibil=+
dn = + tn1rl ,*ro d*
.li.2r d
fi
I ro ,'bo dx(Ya)
t
(YD)
ttI tg @os ra - i sinru) iss --,7
\ln -fi
.r-!:L. il 64-ri q (Yb) t $a) r (\ ) sV-r.f 6r.5..ru ilid, Jru .':
@
f(x) -,l=-@
cn - * 7"rO, e'bo dx ; n=o ,!\ lty.rt... G)
q (Y) -r 0") S|-rl.t3l.ii "r^lL V/ o-rlr ! 1 cJ^i -Lf; [-l,U le! .tf@ Ctt e1.lt?
- ,s f Ji tSVre
G)
f&) =
^V*r,,,
T'
,, = * !-,' flr) , 't T'
dx ; n=oJt\ltY-1 t...
(0)
(?)
\tyqqjtl(v)
C*t'tf(^)
cn- f f", -ib) ;
an=cn-c-n ; bn
c.n=iro"*b,,)
= i(cn - c-n)
.. JJI J'..i
.t.t-j .:-.,)..t b iTF- ) oy,o Jli ;.gb.51 *, c9rJg 61.lrr oat -,U.t
bl = f$) = d . -7s1x1r ; p = lltgb.-lt 6fJ.Li,., t>2j-o .\ Jtb.+tLrt lt stt' Cb q.rl Gr ,r,+ rtt' li dJq r +rit-
cn = + I"n atu dx= + [nor{''n)' ty -+. d, ,(t'itt)x l:"
,"= * " ## yr(vin)r-r'(vin)n 1 = ryr # ( V"
d-ry^i_ # (\)ndo
filr (n) cstrJ-r I t +i \ s
an = cn * c.n = ry (\)"/ ## . # I = ry" #bn - i(cn - c.n) =i ry (\)"/ ## # I = -rn ry' #
lsinhnU.-
1t
;l c,.-"1 *2Vf(x) Cb.:t Gr & ost' ;l-lttf{ - ry. ry :, t # @osnx' nsinru) I
irf u-F-t- t rr,b:.^
sin wr : I @i* - r'i*) ; cos w x = i rtt* + e'i*)
f*[f f .\ ,:; i '.fb (\ ) ^,-.rt Jl-,Sstrr
q.r, s)b_rr .r t+J6.;t ,V_r 1rf )
f(x) = I* t( a(w) '-ib(w) ) ,i* +Y,
f(x) - [* ,@) ein' cl*
a(w) + ib(w)) e'"* J dw
d(w) (a(w) + ib(w)) ; c(w) =' - ib(w))
f(x) ( c(w) ,i* + d(w) e'in'; d,
rl.: rjrL; J_t rs_re a,. tslj 6 b (\ . ) .l (q ),sV-rt
-\Y
@
-T
Irout,-i-f l.
e\c(q)
(\ " )
(\ \ )
c(w) = + f-fA e'i* dx ; d(w) = + [_"fO d* dx
c(w)w -!- f(')rri e-'* dxc(w) =
W _l_ IW) e'-^ (a (\Y)
r:-,1 lyy -[sl --* t (\Y) J l"Li,., o2ye +f(x) u:! JlFl (\ \) J-rt
;,*trt+ d(w) = c(-w) f*rb(a(w) + ib (w)) (\r)c(w) (a(w)-ih(w));c(-w) -\
Y-\
Y
{:
a(w) = c(w) + c(-w), b(w) - i(c(w) - c(-w)) (\f)
.i:.iri-, r.p,e \oye 6-;ll, Cb 5_.*rt JlFl JliC ig L.2ttr,SJr
; lrl <n
; lrl>nf(x) -
Cb.;t JlF.l LLi'., t>sj.e .Y JU,
(ro ) "' JrlJ"r
d/lJtl
.-t;s w *rtt'qbJi u-stJlFl Ji.S.Str.rat*b
c(w) = + [*.\et ,'i* d* = + {7o /* d*
. lo= i ,-i*l'- - stn wlt
lwtt | -n wir
sln wfi , ,tdlt9 =
-wY'
f(') =*{_ ry ,i*d,
JlFf ,r+ o;ye.r)-SlJ4 dl, ..:.-,1 ii Ctr u-sf JlF"l leL;' ru* 6
d,; orLi:-'l (\ f ) AV-r-f :lf(x) usg
a(w)=c(w)+c(-w)=rry
b(w) -- i(c(w) - c(-w)) = .f.r.ii .,1)
f(x)=+frycoswxdw
l1;1."'){:,r,:y r*,,f-.r \ 6i':jr,JkJlj:;l tl itoliC *9 JlF"l .S.fr'
1',,*llr r6;ix = " g-.r: rli
sinwn , r4W=\w
5r-5.+li Jli;u..-*'b' -P F \:
Y, I*lto
OI
g*sinxdx=L:xy
t_1y o)L-.rt'.r t+JtFt ,4_r ... ( rt )
u-l9.r!,-t l .0.\
€,-l' )'_r; ffi.f=or.f ,i_L; Fr{fI r, ri \f(x) eb .rt 'r-,#.J,-r;
\f"(w) c-fu +-t-tt ,./-#rF-r+ .J..rrrr ,i-L; i|r@) r, .rtill ',r
15 b !i r5)-t-.d.'L_1C,:i" srr/ l+:i bf@ Ctt r.f
F"{fI = e I* fttl coswxdx
F" ifl = {1 f i,@) coswx aw = f(x)1t
V} -- i, @) = \,8 f* ftrI sitwx dx' Jt to t l*/ .,r
(l )
(Y)
F" (r)
F', (/) = ftx) - qI- fr(w) sinwx dw
Cbq-rY d*--'l,r#.))[-t;
ff)
.\ Jtl,flx) -
I o <X<O
; x>o{:
i,@) =e f"rorwrutx - ,/T sin wa
,tw
.J*l'.: b
f/w)\ - coswa
w=rE-rt
=q{*
(0)
(?>
g-.rlr .F !'.f Js.tq .gr$lFr t 'tlLif
F, Vt -- q f r @)cos wx dx = q ff6) ,^ ,rl-.* "{ ratsin wx dxl
: -e f(.) + *F, {l}
F, Vt = q f r g)sin wx dr = q ff&) sin wxli rt.*t,-,coswx dxJ
= o - wFs U] = -wF' {fI
il.l rt .I:.-r F.,DL-r+ r-J--r*l- rt-'y",5.i\'ry
F" {af * bSI [af(x) + bg(x)] coswx dx - aF, {fl + bF, {gI
tfusF, {af * b1l = aFs f} + bF' tg}
t;.t,3-t f (x) 5 cryx ^tx tsst J. l4|*r LU, s r>3af$) +tS ,rt -\ 4;i
.6iiix*o a5i ,_":f(x) * o J riq,tLt iUU -f t.Sl.5;
F" {f} = wF" w -fi rt"l
F, V\ - -wF, {J}
qrt o)t_ar r kjrFr ,4_r . ( rA )
,-5c-rti Jljg.,_fu s
(v)
(^)
o'F, {n - F" V} - -w'F" W q f (.) = -w,F, {l}
F, VI - -w'F, W -e f (.)
F, VI - -w'F, {n .e w f(.)
.+t", l:f(x) : e'* ; o>. gb 'r-,#,-Lrr, jt +ai .5J+, .Y
1",ilr (V) I +i !-rr-l :lf (x) = a'f(x) d/-lJtf k'*)' = o' 6*Jtl,(t*llr
* orE-1t
4.T^:i J)
d,=rE
JT
€g.i,_Jr., iri? lt! = f(x) ; -o (x( + - Cb ,r-fu +,_tg.!_+ s +,-s-tr,!rr,
-- ,/ln (q)
r-'ff) =f(x)=# fiud*d* (\ . )
;rJfFl[iU, t otrtr.rL:. il-rti -p ;3h* ee.t6.*b t 6 rf.:,ti.tly.r,
.r:-.1 u_.t!,-}1r+ cSbb .r.lb
(a' + w') F, tft - " rE
F, {e-*I
F {l} = f(w)
It
rffit
6
J f@) e''w dx-OO
( rt ) ... J5t.f'ai
F {e'*'Y =,
: j_,,lln
F {e-*t'S ='
F{af
-1{ax + # f
i, I+k \f@) = e'*' ; a>. gb t-tg.Jrq .f Jt:,
*r'{*'*ir*) 4t
\{Yn
.€ t-4fa+4-)t+(+fJ e rla' "to'
l dx
T
-wla
Tw
I e'*tlla
@
Ie-€
dx
c*t-r [:-. du -- G *(!-l q qi tr
= aF (f) + bF {sl
'* S ,jsf@)*' -l $! c-',J.tx tF lf@ dg eS .Y 4';j.t<ii -r-,tr y-!fJ:r l;'i,h'f (x) ;:i ,1
iwF{fi (\ \ )
fl-b, .*)LJl
# tVr,totu dx=# [f&),-inll,,, $av'* d,]--iwF {f}
F Vt - -r'F {fi
F{xe-"} = F { (t'" )'l
(.f . s) (x) - tr (x) = Ihtl s& - t1'dr = [7W - t) s(t) dt
c-;f {*t jli ,_l J.&::.r
(l Y)
.+q, lro-" Cb qry.,l,_r+ .f Jtlr
( rc'" )'I-t=.i FY
-\Y
*' n'=- I iwF{e'"}--f i'}"-;- -4e'Ty y ,/t yfi"{:ji crl "'! jl# .r:,-,1 lrv ri;.4 l=..ii 1 o-.r/ ,}.r+:lt ;L-l ..st-ui jr,-1(1
.r'sbar-e r,+i *,,fu-.r- q
€C -L-y: i,ii? rg-r.r.r,o &_*f.g t 6l:g Sfglj,,:1n4
(\ r)-€
J,-r" tttF{f} r{gt .t<ii r-iL s sf ,-t!.i$-ry qj qr{g} tFVI q.rt".aiJ+q.=t fi t .-at ;l jli: u. U g;r;,1 ..:,-,1# (f . g) t:!
,.-ptt-:J.-u -f tcsl^* +*,sg$) rf@ +s,rt.(.',t.t', d:"at)f 4-at
F {f rs} = ,fr; F{n Fkt
FV.B) =# f _fn,ls@-t)s-i,,,xildx
I* tTtrl s6 - t) s-i',, dx dt
.t<ji r.- lLr tF I .'4l-3s;t 'tiilo,, s tlx-f
..)LJf
\-- {rn
( tt Jrl J,..i
FV.8) = l-r,/ln -J.tI ft,1 g@) e-in(t + u) dt du
g1u) e"i* du= # I-nq e'i* dr
=ffi FV:. r klC*t-.-r 6y rss -; #-h:lrs$,-,'-€*,,-1,-r+ sE: p t1t ,i: FVI Fiu.
@
(f . g $) = f_fitr'lr E@) ,i* d*
..:*,1 jf .itc^:- !.tYrt-, JiL-, ,F elt gV J-r;6.t>J'l {jbl ,-,t5Cr_l .SkllJ) Cli jl dt +,;g.i)t_"r; ,gly. }s+
0J.,3 J- Jit-, .t.\.+1".t lt -tj-;|.:.p, t Cb u--g 6f .\ Jtl,
OI-00
lLn*, ; -,t(r(cf(x)-ll
1:n-X i o<X<rt'Y
Jort = o \1.-,1 rt Clt 1.3_l u. j:
bn=L I"rinnxdx- Y pos!-cosrut)7V 'n tuft Y;
OrlJti
f(x)-L > I1co, nI -cosnn)sinrur, Jt ,r:l n' Y
Cb.-rt 6r .Y Jt:,
+ttb
o^=* (l f *" +x)cosnxdx+ !."riz-x)cosnxdx
o, = * V + sin ru . ry + 4 cos ruJ.o'
* f +sinru ry-)cosral.nl =fif\ + (-\)"*')
o. -+ r I,+" +x)dx + li+-odrt = .
f:lr .:,.,1 6rj 6b O.;a
bn= o
etl.;atti
f&)=+ f * k\)n*,+\)costu1l ,r: r
.+tr lrf(x) = x + si^nx , -trt<x<lt Cb t t 6r 'f J[le
f(-x)=-f(x) i an= o
b,= + !."@ + siru)sinradx = fr f !1 t*nxdx + ["ri*sinrudx]
b,=r+jl- fnrin'rdx=Y7to
crrtrQ
f(x) -Ysinr + Y; + sinrun=l n
cb *rt tsr .f Jtb
(fr) .... Jrl J.ei
; p=l
.+tt* b
an : !' rro, nn x dx + J'(r - x) cos n tafla - # kV" - l)
a. 3 o, bn = !' r rin nn xdx + J'(r - x) sin nn xdx= + kV"*' + \)
f(x) = i , # (-\f'- r, *r)+ J ('v"*'- \) sinnnxln=l
(' '' ocx<\f(x) = {Lt-t ; \<xcY
.+t t ltf&) - sf ; -*1r1o Cb t-t-f 6r .0 Jtl,
f,e- cos nx ax - \ , dsin nx;t "
s' sin ru l" 1 , -e'cos nx
" i-"-;t "
", + 1"t 'u^*1
": + InT'o'* '*l
.ttott: t J
7t -il
ln . n
| + t Je*cosnxtultl-" n -.t
-J-{xt
sinh nlt
d/lJt+
f(x) * 3 +({ @os ra - n sin ra)l,r=! \ *n'
fr.-:i.2)
(to) "''Jrl Ji''i
,sT r,.,.+7;trrr.:t"r \f&) =.rt , -l<x</ Cb t'-s! 6f 't4-
v))te ,? Jlb..r*.rric--,)1b \ . + . + +... r \ - + . +-+... G)-v s!-ril4. -p t-)w
o^=* !'r'ro,ffxdx=+ r* !',,uT*t
- r t ,ror y l'= I'', (-\)n I ,totut tutt , lo flfr
a.=+{tt,d*=+
f(x) = x' = + . #,i ) r- \)n cos +,,.*t-fx =ntl=n f-r) it*d.lt;,..r-l t: ft,SV
n'=o' * i il-r.;, n'
'*+.+*+*...=l,_ \ + \ _ \ + _ :o't-T-F -"-
\Y
f;l:x = o tl = n ,*^) )lj Jl JV
.+tt \tf@ = sinnx i o(r( \, P = \ Cb +s!Gr.V Jt:"
ll
an: I I sinnxcoslwcxdx= I [tU(ln * \)ntx-sin(ln- l)nx] fucc
-Y\-X-It fn'- \
-Y1t
f(x) -Y_fIt rtf I cosnnx,r=t Ynt- \
.r.rtt \f(x) = x' ,' -Y<.r<Y Cb q':g 6f ..^ Jt!
Jan= o
I
br= [?.nfix'sin"" xdx=[
Y
Yxr ,ot l'cos-lrun Y l.+ l-- !.'r, ,o, T *,
- \t(-\)"+' + ? I_ I f"rinnnx dxl= \2(-\)n+'fun ntrtr ' rut -o Y ' farc
bn -- (- \)"*' \?n'n'- qtn'trt'
f(x)\?n'n' - \?
n'trtt
.r-tf* U f@ = e'',' o(x(Y't Ctt +'-s! 6r.q JU'
ttle-' cos w dx = * [ e't ('costu + nsin w].'o
n(n' + \) '
q2(- \)"+'ntfit
d/-lJttl
. nfixsut
-Y
6sL.
n=I
an=I1,a
(- \)"*'
(rv) Jrl J.ai
=1\-e'tn )+ : a.=\'e''o' fi n'+\ rt
bn = + !'n ro rin w d^x, = # [ e-' (-sin ru * n cos ta)],'o
- \-t'to . ,,,
rt nt+\
d/-lJLt
e-, = \-e''o t (I+ I cosx++ coslx+I cosyx+...) + (Is;ru +{fi , y y-- .O'""'-
\o ,y 0
sintx + -!- sinYx + ...) )\"
CV g-r+r,*rJ cgr .\ " Jtb
.+k b
bn - r,{ f@ sin nn x dx
=r[ {',* -x)sinnnxdx+ !,' O-{f sinnnxdv)
T
[I-, , "o.ir(x)=tJ+;|o.r
er_;y O)L_.rj, tdtFt.t+_r . ( tl I
- \ cos nfix=r[-(=-x)-Y tuft ffii+rt'$-irry
*+ant = fi t\+(-r)n/
n*Y
Sln fUfX ,rTI,
T
=#t\-(\rt -Watfttr
I@) = ( +- ]t'i,,'** (* . # ) siny'o* (*- # ) sinutx *...
.+k ttf@ = srnfr ; o 1x1z Cb .friJ u_sf 6f .\ \ Jtlr
o^ = * fnrinr, cos ,x d.r - * , Y [; !.nr^yx sin nx dx J
-? , 'cosYx cos nx l"-,*I-- I.
Ion=fu(t+1-1'1',l ;
or=oi o.=*
fk) =-L-* f ?e' '\ ' Yn 'nj,
, ,*, fu (\ +(- \)") cos ru
.LtSi .t-.r: u.ftx) = coslu t-lt<x<o Cb t;! 6flab, :-t22,a.lI J['lr
+ 4onn'
crr = + I" costu ,'ittr fla - { t e-bo sinta l":
* !" rinh re'bu d, I
(ff ) ..... Jrl Jt.i
= (- L)" sinh n + + [ e -bo cos lr;l" *, !
or -ro cosla dxJ,t llt' l_" :n
= (-\)n_ sinhn + in(-l)n cosh n 'ntcn
It ; -tt(.r(of(x) = {
[cosr ; .1x1nr
cn=#sinhn.#incoshn
f(x) =*,i_ frf,, [sinhn + incoshn ]e'bo
Cb.;y GrlL;, c.,.tt' .\f JIL
.r.ttd b
cn = * 7" r,r) e-i'o dx - + I" r'uo dx + + I.n rour'bo d,
= *e'bo l;* [.n@k +e'o)r-budx
: 't k'in,t-\)+ ' [\ riz(vn)- \ + \ ,'itt(t+n)- \ /Ynn ln 'n-\ n-\ n*\ n*\'
= i (\_(-\)n)+ i t(-\)"*'_ \ *(-\)"*'_ \ rlnn Yn' n-\ n-\ "+\ -n+rt
r-.r-rr st,.!/.r t+JtFl ,V_r .. ( O. )
;=#(\-(-\)n)+# # |,*.,\
c,=I f e'bdx+ \ lnroue-bdx= l+Jnfln :n ln -o
n"ou dx - \
Y
\L.--
ln
{,'
rSrtt
I"d, * t'tl o
f(x)- +. (L+Lv+ \ ;' Jt f ln rt=_@
n*o,\
Y
JT
fJT
n r3Z/
Y fr n:,
; -fi(I(c
; " 1X1Tt
{Ly # rcv"*,_\)rp
+;! 6r:te$t-3s;t tt .\f Jt!
; -tt (-f,( o
; " 1X17t
3 sin(rn- t)x- - Jl*t -n: r Ytt- \
, g-$c dr*,b JlFl *-b;l l-r;,1 .*t ltl Cb u-sg 6r +
3 sin(Yn- \)xz- -!y61 ax:n=r Yn-\
S cos(Yn-\)x [-x ; -nt(r( o
;, -lIn-lf - l, ; .1x11t
:.r-lrtr , = + ,.*t-.r -n l:n {trti .,, il g.sJ d"hjl .Jr$lFl ti
cos(ln-\)x -[-x ; -nt('t( o
-@-1r i o.)c1n
(Or I "' JrlJ".ei
J lJU ltt ot))y .7Jl::. J dl4t-; +>t;--t ['n, n]d^rU i8 sf +'S ,f t'\0 Jt:'
: rf -rJ.:,tj .63i r;!g sf u-t-f --l-r-:ii u. (cn , d) s @n , bn)
* irr dx - a{" +;i, @nbu * c&,,)
v-l-r-Bn sAno6y'.c-,,1 u-t! 6r csl.rl:h Cb.t<ji h = f+8 €,?l ^+tt": g,*rl: .KiT rJb Cb ,.r-l u-s!
An = L fo cos ru dx - r 1
"7 ru, ra dx+ ] ["g ro, ru dxfr--n v ln fr'n
t Brr=bn+dn
d/-lJt {
s-J+-t
(\. ,-f-F) 6t-5.:"tJ .lly r-l J:
Arr=arr*c,
A.:a.*c.
+ y'dx =* * ,i, @', + b'n)
! +i b. sh sB sf ,3,l1Jlr-.1t, 6-rti;l o:tn:-.,| ! c,.-,'l (f y Jlrrt, 6sV t 15
* io + s)' na = *-it'*
,rot-,r (f + il' = f ' + 8' + lfg:t-;l
*+_{"r'dx++{fs*
r,_.ry J)L-.p , tdtJ:t ,4-r . ... ( OV I
+ s)' dx-+ Ii' ar - L II'*
- (a '*c ')' *i ,l
,o^*c,,)' * (b,,+d,)' I =-*,,:,, 'ntb, n)-* if 'n* d, r)Y' il=!
I f Is,b: t, I'(.ffr-nfr-n
=++-.'.-*i1o'YY
= {:
n*b' n) . : ,(c'
n+ d' n) * a.c.
* ,i ,(a,rc,,
+ b,dn) + if 'n*b'n)
fxrtx : o,l' + I (o,,c,, + b,d,,)r(t Y
vFc'. s-Y t=l
(c'n+ d'rr)
\f7t -fi
cb .\P Jtl,f(x)
i o 1Y17tr
; x>Jt
I
dx:L I,T
o
-Y Ixt
,tsin wx dx
.+5.=-"t^, t-l
-cos wx t ,t
-
].w
=LI,U
JlF"f ,i t ^-r, ! r -r..--r: utl ,-f _r1-' JlFl S-- *2j,,o t'.12
\ - cosnx' ''\'t"e' sinm dxx
b(w) [(x) sin wx
(OrI "' JrlJ"'i
Y \'cos wn=-ltwd/ltq
@
f@ =+ t rysinwxdxo
.f-:lrx=n 6,1;1 15
1* t-tff*" sinwn dw: i tn"') + f(n*11 = ?
@r \-cosxtx . , 7v|
-
SInfiX (17; - -"xY
Cb *rt .Jr,ry .\V Jt:,
r$..-,t-., rr f ryax ;:i n'. ?i ! -r +! r-r
itnl -F V\ -- i, !Jo,r'iwx6= # {,' o*'* = #,; : rffi ; w * -
Ir ; I'l<tf(x) - {
L. ; Itl>\
c-furl,-r., J-r;;l ortn:.,,| r.r Jt-i tO =Qf;l: w-- o cl)l y s
q.rys*,.r'irtdtFt,V_r ... (of )
f(x) 7@l dn d*
fl-llr
Irl<tlrl>t
t-rJ r-l 4rr;ir= . Jrlr .llj t:
@\r--
I,Iln :-
\ i rt*"- do dn=l t ;
rn :- w [" ;
"- sinw|
-Clw=ltl-w
O.stnw . fidW =-wYt-r I
{:-{
n {# y = $w)" F{u|.S.r+r.rLsn ,*.J' tb fcsl;l q .\A Jtlr
f*rb, Cb rr cSUl .{
F{uI =* I ntiodx,F{u'l= iwF{ul{Yrt -a
,.5-tt.rlj.r l,i:.,| ! t F{u'l = iw F{u'y = (iw)tF{z}
F{u{0y =, 1iw)n Ftu }
.f .rr., JLs .\ I Jtb
O coswx+wsmwx, r
F-w=
X1o
t *-a
i Xlo
l" i x1cf(x) - {
lot'' i x) o
t-e'
JJI J..i(00 )
a(w)
p*rrr ! .ri r.,-.ly JlF"l t g#,;Jb:t l.r
-or'' cos wx lss - t # (coswx * w sinwx) l.* = ,,#
a(w)=#
b(w) = * f.-or' sh wx f,a - t W (shwx - w cos wx) ] .' = #
f-rb t+,i;lr
{" i x1o
i,W+ # sinwx)dw= + tf(x+) + f(x-)t = 1 + | x=.
lnr-' i x7 o
.rtJr-J-t.V.\
.J.i:..r dll* u3. o.l^i crll 6LrrLaU i J-s Cl-r;l .5='.l-t5 '\
[r;.(.r(t [.; -\<l<'a)f(x)= { : rcr(y
b)f(x)= 1ri.(r(\[. ,' \(r(y L;
[r ,' .s.r(\ [t-t,' -\sr<Yc)f(x)= { d)f(x)= 1x
L"' ; \<x<Y ty* ,' Y<rsr
.J::..r )/ ,u I g$c b- trri ,ei -$ gly.ll '5.11-tf 'Y
a) r*lxt +rr' , xLru, + , shla , d , elxl
b) f(x)
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-t t:-.r-,1le rjti rl-s.l .r)'dU 6st ii fti esi 2, o3}v € et +tt? ':*'l
;1llM J* t3; o;l$l .Kri .r-,!p tltx rt'e; ;: ,J-p r-lr J. Ji dr,,l o;UulJl
U c,,.-,1 ,,.1;. >rSu. ii ,5;-:-- Sg ra l5r^. Ssi d/-ltt+ "ry J'rlf pLx
F=Trsinp-T,sina+pLx
y.lr1 Srr,-.,-,[:-i J) f-.^? l:+ ?-*,- ! r.f Jl$, Ui;rr .ly$ * C.h )l
c*t-r d/ltl+ c-''l
T,,sin B - T, sin a + pL, = PM *fjg q+ T Y.6sV u-l 'h'-h F -" t'l
tsp-ts'* +Ar= t *#?--t' tlt 4i *.tg P stg a r(!-l u. *i tr.r Ar J it 6sV d.b f=j; l'l
;i;il.ort-,^!olsh, ..(Vf)
, Cot-rf .:*,lr+ M s, !l.E 1r .J/t , brU; dbrJlj\ / \u(x+Lx,t) _ Au&t) t +L _ E_ 0'uLx' ax ax t 'T- r at,
f:l: ItJ'-, j,.a g^.. r, Ax ^+t+ J[.-d'L*.p-=e-ArrTT
d'u , 0'u D
-=_ -Att dtY p
,T,'- ; a..f tplT J.G:* U-l d-h.r)-Jf.r-it tt
(\ )
.rr d r(!-l t q.9; t{ .t*T &-tU g(x) s f&) f^.";i q ^Jrl
\>er.,,.r jLi-rl .6y'
.f-,rlr c-^-i ,/b;l LSblx-/ Jx=. -btii
u(x, ")=f(x) ,. o (x( \
u{4.)=g(x) ,' o (x( \
u(",t)=" i t2"u(l,t)=" ] 12"
.-u!"f (\ ) d:--, ,sU '*-E r;r.j ,.51 (0) tr (y) h,_trJ q (\ ) d.rt.^,
0'uaf
(Y)
ff)(r)
(0)
Irt=ctltnr ; "1x11 ,tlo
u(x, ") =f(x) ; " SxSl ,
ut6,")=g@) ; "lxSl ,
tt( ",t)= " i t2 "
u(lt)=" ; t)"
€,y.1- l-, .,p l:* s y_tbtef u,"V./t- tr (\ ) &^-, J- q j9:Sl f"(ta1
(tu1
(tc1
fi41
fie1
.#,iJ Jst \ ,-;-r- ,-r-u +I".', d/l J- cSl;' s C*b .f (Y ) d:.-, b dt-, o,,-l
.{*5 6' $lJl.:,-,,| g_r lu ,F :.sjLJ& r: -I;l 6t{-r)^r Jev 15 t, trr.,c..
rlrL. J- .St* u. e- €C o.rLri,l =lf "f-l,S-
s s:+sc _*at-;l rL--, ,F Sl-t
'elf €'l F-S d:t^, grl tSly. u.2--c :|f ,5- 1 tl*r: J 'JS.Jb 1r (ya)
GF'
(v0 ) lJt J'''i
df_+.-r:fi.r., (Ye) b (Yb) Lrly' tr dg.:t=: 6:* tr (Ya) JrL' 6ltt 4te(Y) {L.-, Vlf t{j (\) +L--' 4;lt:lf rJr;a bSw- +'.tqi ! Vlr+,i'l 6":'-'l
.ry Jelli
d:-, LS,lJ.uQc,t)=F(x) G(t) os-t' \ Alf 6F q (Y) &'-' S- elt'
tr -r5i-r- d:..-.1, l#i ,-tr.hl."V dCot-.f ,S:-P ltG(t) s F(x) ..r'r'- g;l: itl.C*t-f nJ:L, f us=F'G 5 tto=FG s u=FG
'"&VFG=c'F'G
t+i;l-r
-=F c'GgsJ.a-,,| t ;l u-ls kJ;r, i* sx .tl.r^'tt tr' 6rb ir/-l JJI i* l(.,-l + -h
jl ..51 ,s 6.r-rtr :rt r,rk sc:\. "r:'ti
r'rr 6J''t't i.tl $li€ uel La; ot
f..i! c--lb s#- -rJ! rrlrr .li ! .51U ,.rrl.rr \t+Y 6k;-;
€)d * ,L.;{Ar d:L' J) q l+i -rl-r
F'-kF=o , G-l<c'G="
f-rl: $b) b; !+i q Jb
u( .,t) =F( .)G (t) = ",
,f-f\6t-rrlIt = o crc-{ elru.J4rC(t) ='lJiG(t)*" 6">-"l.te-{
b;, ,,r. * jt JJ.t+..^ F(.)= o +f ..a-,,1 ;j) &lJt+ e).ll) "AV
d.ll ./bll tftl
lt fid:.-, JJ:SI f, F(l)= " .,, rl) u(l,t)= "
F'-kF=.; F(")=" , F(l)=.
6F,=o ,1,l, .t<ji k=o o6y' .f.^rrr :l-l ,ftJ. )t-f k ,-l)$- .r.)u, gl;l q
C*t-f F(t)-" s F(')=' \\i \+-f b 'c'-'lF(x)=ax+b 'l-*:lTSblt
F'GIF c\G
jh ,ili;-, ! i))rl,r,. . ( yt )
d/-fJq.\:-,1 di. .- ,Slt rs+t-** =lr t;e s r.l .jy; 6u=. l".'i 21 5F=.
.t:-r-i+k,.,, Slt elf , pk=oJy-e .1F'-kF=. rJrL-, ,Jr* :\*.t<ri ls-ptlo €C ,_i-r, JVt;T;t t a=. ,rrl, F(")=. Li u.*j tr.-r-,lr.f F(x)=acoshqr + bsinhpx
P ,-ilu-,l. ss fl slt i-n sbsinhpl- o .r.l)d qdi F(l)= " L;F(x):bsinhpx
9p J{)Li, tsj4:-.4 {J:-, Ji- Jt<rl k" ..rtrl+ F=o t+i;l Jb:" d/_lJl.l .r::*r
,sF'+p'F=. ,r.r-.ll)
t+ri;r sk=-p'<. {r:rrt )l;i ).,b, u-l cll..:,.,,| k cltf*t-.r F( ")= " ,.if t .+i ! .c^.'t F(x)--Acospx*Bsinpx ,t_r* =lr .,bl)
,SlJrBsinpl=. f-tsin r]-,,) +,F(l)=" ,-fljtf Jt- F=Bsinpx t+i;l 5A=o
St* +q b p ,t-ltb. t B* o .t5.1-,1 ;j) .-L-t{ F* " :lf L5bl, d:--, r(!-l
sinpl=o .;h.ir n g:..o)Jry' 6ljl q tp= T *ulLsinpl=. l5rf .-,1;sl
\ alr q t Ji 11 5JJ,l;i,a JIJB)IJi. c.^-i i.lY ixrrQ 5 J'i JolJt -F jlOy*a
Fn@) = Bsin +,)L,n= ,,Mcnl.: rlj 1 G-kcrc= o rJ.:L.. f k=- + ;xf y;(,,_* tr p-r,.r
.f J,lt riz 6lt lj +JrL, q J-.L- {.F.;i J)
,-/-t* =lr Lgbl) rJrL, ;11
G"(t) = A'coslvnt+B' sitt),n t
un6,t) = F ng)G n0' (a ncos),nt + b *siil,nt) sin ff x
G+)rtnG=oi In=r+
t+i.;13 r:...,1
bn= B' B , a'n= e' g;si -lt rf
(vv ) lJr J''i
r.a-,1 (le) s$d),(la) ,J:- ..l-r+ n uJ )Jr y',Sl-rl q un(nt) r(!-l ,'. p
;.*t-t>,^,tl &.-, )l ,llf t+i U-+,. d/-lJt''t
u(Lt)=":, un$,t)= : ,@ncos)',,t+b,;in)'il)sin + "
(?)
3 an sin { r:1r1n=r I
Cb.;t '/H 4 flJ6sV J2t;1" r5,.:*'l pY e-lL;;rlrl't; tl;tt 6;11
f@ eV u-s! d-* L.4'=;re At ls a,,'>-"1it5 )pz u-l eV: s\f(x)
u-t'€rt+;l
)l ,?-(Y) &'- rd, \V ilbn -tan $f oat-' sli,lra '*''"1 d:"'' )l g-lr i
gil: u(*")=f(x) b; u.*i !.C=5f oti:"'l (lc) t (lb1 u'1 '
(v)
f*frf u,(vr.) -g(x) ;l o-rLc-,'l \ : (r) d-h ;l 6, j';;:' ! Jt-
ulx,.)=:, bn trn ti, + x- g@)
eb.-.rt ,-/H L..| .,,-r. l.ll.l.tb,rltn c-,,1 ils,S-lV grl .l:; i-tt'Slt s
b,=*{ s@)sinff*
an= +{' 1'1'*ff*
t+;i rtr g,5-rt+;l S&)
(t )
jh3rr-o^!&)rtr,. .(VA)
J (v) 6u ilbn lan '*lr).li;-r .5,:*,1 (?) es-* \,llrL5bl) (y) 4L--, d/-rJtl
.U_iCr.t-r,: t (A)
.,*,-lc;L ,!_b.r t otJ.)l{k Lb.,,l +. L-,1 )l t,JC, ;f!, l)t J* 1 "f, .\ Jtl,
u(x, ")=i o<x<lll
-x); llt<x<l
.:.-l -e tlt ,'lb-rl lj rz,cr-' 2
ur(x, ")= "
,l-lt\an= bn= o
f+,l*n
+t'11'1'*ff*'
- hk nltan=ffit-T
u(4t)= # 2,{r,r, r{r^l,,,tsin+,
! Alf ga;'a t +L?.*, y-l J- cSlf .f-, Jgl
1.*t-f \.\ ":;- ? JU tr-
g,--rl:(2) t qi L. iral;l
1f I *L-, t ir-r+, Jt-
,->yr.O
(vr )
u(x,t)= v(x",t) + w(x,t)
.:-,1 ;lS 2.JLz Al Sl.r,v(l,t)= "
w(",t)-p(t), w(l"t) - q(t)
irr rl'ai
(q)
5 v(.,t)=. df=r\r;r tst*ltw(x,t) s 1-sltld.r.lL pA:..- *lr d1,ot* st*\.,w(x,t)
ott' ^t,;.lf LS* +JliC d:'-' U-l Slt C.lf CPt- 4;lr4st>"'l
w(tc,t) = ax*b
b;' 6tt )l 5b=p(t) Fb-,,r' w("t)=p(t) b; q qi b 'c;bv
w(I,t) - q(t)
rlcq+(q(t)'p(t))
(q(t)-p(t))x+p(t)
,it.lJt {
t+T;l.r
(\ " )
f;ir.1,*t-.-f 1.,,v t b*t d:'-' Jt-
-b rr-r:1r5,- l.6s i* J:.c-r :lr 4,,sbl't{r dL- grl 6..:'"'l '+'{
u(xt)= v(+r)+ + @|-p\)x+P(t)
xItt:vil* =t Q-ir) *i) , tts:vs
\o=T
w(x,t)= +
vu * i O-fl *P-ct vo =F(x,t)
f:i Cs"') q (\a) 1r ..li d)-sCt;dt- t'1
..li 1.l 6vu ' c' vo =F r(r.'t) le.,i ;l r
.g*tr..r (\. ) r (\b) b;' q qi t{
F,(x,t) = F(x,t) - j O-f\-it
u(ry. ) :v(\ . ) + | tat " l-p( " ))x+p( " ) =f(x)
,)7 ot;* ! oysr,.
ol .2r 6v(+ ")=f ,(x) VT jt .s
p*t-.r (\. ) J (\c) 6s1 jlr,Ja..l
il -;r l5v,(f ")=9,(x) q;: -2:
f)'fv LglJy; rL.-, t d/-lJt+
ffa1
(rn1
(vc1
(ra1
(re1
f ,(x)=f(x)- | re r "l-p(.))x-p(.)
u,(x., . ) =v,(x,.) * + (Q(. )-p(. ))x+it( . ) =g@)
8,6) =s&) - | faf ")-it(. ))x-it(.)
vs-crvo=F r(xrt); o 1sc1l, t) .
v(+ ")=f ,(x); " sx_{
vr(x, ")=gr&); " sx9l"
v(.,t) = oi t2 o
v(l,t) = oi /l o
(\ I )
C*L.r ffa);r .ri.n.,SdrrJ&b ! .g:l t1^;.ff) d*, 6ry
! (G, + )r'n Gnlsin +, = F, (x,t); )"n=s &I
Q) Cst ;l ;.t\i ut b O) dL, .:l-l+ .t<iT ttlS .j,-o lllF,(x,t) 6,1a,)c_.r: +#La
" ot(;iI ,({J)* " -sl JL .+.".r (\ ) 4.b"', :rr t (\. ) + +i ! -r ,.r.ri d--,r r,r
OS2,e n C.lf 61,4,:--4
€v(+t)= )
,t- I
Gn\)sin +,
u*-Sf F'(Xt) Clt +ug ,/_r+ ,.-.ta.,e ;.lra Jlj6lrcn+1'n Grl 4lr4;.
EA
:.J(\Y)
llrv/ ,': I r I -U ll
d (\ \ ) ii
O;ya u.r-ltu g.\f e, +J:Lr J/-l J- t,1
(\r)
a .+i 1r.gr."t.-f r,C-.,| (\ Y) rJ:L, 't'td.:ll+
,*-G'n1t) oT.,1: r5
.t_l C )t JJ Oy,e
v,(xrt) =0ct
,t: I
(u,, cos i,, t + b,, sin ),n t + G'n611sin + ,
f::t-..f 4 j q fc).r (\.b) -r d-rb;r-l q .+i tt
* G',, ('))::h +t = f,(x)t'f.r.'1=2it,,,J= 1
!'1(.r, o)= I f,l"
arr= 'Gt ,r( o ) +
* G',, ( "))sinnltT*=
nill*+ I'.
8,&)
F,(x)sin
l+,1 jl-l
(\ 0)
t(\ r)u^= t [-G',,(.)* + [.t g,(x)sin T *t
--l-e !(t f ) J:'t;lu .li.r) 6.:*,1 (\ . ) o2j.e ! ,jr dbl) (\ ) d:.-, d/lJt l.!-1./ \:-e,) q (\ t) r (\ 0)
y; +L--, J-.1,1 ?tb.Y Jt!u,r-;to=li o<.f<\ ,/)o
u(\.)=x; .(x(\
jh clr' ' ' ! diYrtl ... ( AY )
ur(rr")=Y,' .(xS\u(",t)=lt; ,>.u(\,t)=t ; t>.
5 v(",t)=o l5,.*L-.f ,S.t* l.tw 7u(x,t) = v(nt)+w(x,t), r'=nr"f 2lS ;t a
C*t-.f dt,frL.2w(x,t)--ax*b e1...J,,l )jbt c-l Gltv(\,t)="
r+i;r ,w:-tx*yt cr-lrtr o: + - -t , b=tt\
u(\t)=v(x"t) - bc+lt
r(!-l t .+.1; tt
v(x, ")=u(\ ")=x , un=vn , uu:v,
tvlx, ")=uly", ") *x-1=y
.),-l d i ts,->.22-a t Y 1 b*td:--,
vu-v*=t , v(xr")=x, vft")=x, v("rt)=" , v(\rr)="
@ gjlrf,c/ J) J)-te \ C.lf LS,t4;"'a e., rL'-' U-t )- Sltv(x,t)
1l*t-f eJrL, .,,t .li .lrl: -2lj L
|,rr"+n'n' G)sin nnx =t
t+l jl-r' = Yl (t +1- \)"*')oflfi
tGn=an cosnnt * bn sin nnt + + (t +1- \)'*')
l-e..Jil)
I
Gn+n'fi' G, = yt I ti" nnxdx= lL cosnfi,xnlv
( Af ) .... lJt ef..i
@
v(x,t)= t I an cosnnt*b,, sin nnt+ # (t +1- \)"*')] sirurx
p*tr.f v(r' ')=x r3-l q qi q
:, a,, sinn nx =x
t+,T;l-rI
an= | I x sin nnxdx= -I- 1-\)'*'nfi
.C*t-..f v{x' ")=x tSss ilv,(x,.)=,,i, I nnb,, . # (t +1- \)"*')J sin nsrx--x
d/-lJt{
nnbn* *(t+1-\)'*')- y{ r sin nnxdx= fi C\)'*'
b^: #((-r)"-r)+ #(-\)"*'
U-1
4+::j J)
u(lt) ---bc+yr+ ; t *(- \)n*'\ cosnntU # ft V"-\),n=t nfi I'
* 4 (\)"*'l ,in nnt * # (r +1- \)n*') \ sin ntu
,p l:-'-.Jr *:.-l t.r-lb.f Jtf'uu-l Nr'=.r* lg u ,' o <.r< \ , l> o
u(+.)= (+ -n/t)x',' oS,rS\
,)y oliz:.t ! 9)rter (^f )
u{xr")= \*r 1'
uy(.,t)= !;ur(\,t)= \ i
.SxS\
t>,t>.
\zYf
u(Xt)=v(nt) +w(x,t)
.2lrt ;rrq eli C )tb, 4l Lslr4vy( ",t) =vr( \,
f{rlt owt:b d/-l q qi !.,' =
wr(.,t)= t=u
u(\t) =v(1't)+( I - 4 l.r' +Y Y'
vo-ldvo= Y|+r ''
v(x,.)= - t ", v{x,.)= \ *r,'
vr("rt)=o , vx (\,t)=.
vn = Y|Y.rr ; vr(.,t)=. , vx (\,t)="
a..lrr;. ,li
t)=o 6pot .-r ethl.rw(x't) JV
lax+b t+,i ,t 6 w=ax' +bx f-^)
d/-ltlq wr(\,t)=la*b= \ J
4-.,-l] r5
lt-xY
t in=Y6v.rcr q {+il* .f"5.f ,F F(r) J"r.t Vv(;.'t) -F(x)G(t)6;'z 5.(c o ',
f{Jl'v=FG
\\i ) i-f J.rL- t r+j r
f' -KF=o ] F' 1.1=. , F'(\)="
( ^o
) .... lrr Fi
'{ F' (' ) ='
. ff;. ";;=:',*" l,' o'==
*J! S ;, ;,.,::
'1.1b
k--p'< " gul q
F'+p'F=. ;F'(.)-' , F'(\)=.
C.-,,1 ll ,-lS* Vlf g;l.2lr r5
F(x) --Acospx+ Bsinpx
,',*rlr ;si ,g,sr;t 5
F' (r) -- -Apsinpx + Bpcospx
cr-ftur B:o \:Bp:. sJ)sfq+ F' (")-. b; q.?i tl
F' (r) --Apsinpx
l.sA iljgp:nnt -.,till ! rSF' (\)=-Apsinp=. .,;b F' (\)=. +(!-l +; )r:
*l-* gl-rlr F Cb u. by.-,- +lj-.., 6 -i; q+ 1 );r.r ..rtisl F" -,iJf,;t,2j.o u.d.l-r. LS# q Jt.>.o*,1 F"(x)=Acosn"ttx
v (x,t )-,,:. G,,(t ) cosnnx
* f*:t-f rjl'...' d:L,;t ,.li CJ, ! .y:bL-t'
: (dn+Yon' n' G,r)cos wrx -Yo+ x
t+i;l-r
Gn+t6n'n'Gr=r{ lro*rr, os rurxdx:l I.'o^nntdt= # K-l)'-\);n*.
d-*:lr 6!11 15
G,,(t)--a,,cose nnt *b,,smo nnt + # ((-\)"-\) in*o
1r:lr /t= o rsbl q ..:-,1
u)y ati* ! ti)rl.r . ( At )
Q.- !.fi a+r)dr= o \
c-lf :lf LSblr aS
G.= $r'ta.t*b.d/lJq r,.]-,i
v(x,t)= *r' *a.t+u.+ilancos6rant+bn sindnnt+ #((- t)'-t )l cosnnx
€v(x,")- b.* t, [an+ #F (1"-l)] cosnn x = - +,
b.= !.\(- itl dx= - +
o^= # kl)i'*'* r)-r{' rrornntdt= # Gl)"+'+ \) ( # * +)
@
vlx,")=a.+X Atflbn cosnfix= \*x
o.: fo(l+x1ax=+
Ntnbn= y.f it*n; cosru( xdx= #((-\)'-\)o
Yb,=#kV"-\)r-l.,-\
u(x,t):(\lr - n/r)x'+ ar* + (+ l' +rt - i I
*,i, t# ((\)"*'+\) (# + +) cosvtnt
. # kV"-t) sinonnt+ #;o rcV"-\)I cosn^tu
e-f d-:L, J.4)ll .'!-- .f .Y
d:.-,
It11=ctlts ; o1x1l , t)o (Ya)
u(+")=f(x) ; ulx.,o)=o, oSx(l (tb)
u(",l)=o ; u(l,t)=" , 12. (tc)
dbl di.-., d/l J- cSltl.t,fst lix",-l -;r r1.i J- f d.t' * ]Ut'.tsli;u. 12
.# -# ,-ts.t d/-l q d:.-, ,F Slt.c-,'l lyyr.r)l: ')u r, +5 €rt
z=x-cl J v=x*ct
C*t-f -P r:.' d/-l 9 +i tr r 6^)./rtr:urv,*u&t= c(u, - ur)
t"yi;l-r
uo=c{(u, - ur), v,* (u, - ur)rzl =c' [u* - lu**uoJ
l&+*sIto=u**lttur*uu
++ti .ri ;l 6$\rful ! .l uvz=. f*U-F &..- rJ.ll^. t, teT Jt-S O:-&t- tr
f,rl: rJ:.r G, I-Ssl\ s ur=h(v) ty'u,
,= I he)dv + !t(z):Q e)+rlt(t)
1l .:.-,1 e2V $a) rJ:L, elf Oallt ':
I
jh *\i-:. ! e)rhe ..(1,t1
u(6t) =Q 1x+ct) +1p (x-ct)
QO+ !t(x) = f(x)
c(Q' &) - !t' (r)) =.
Q' $) - tlt' (r) = .
Qt*l-qt(r)=*
Qat=+ tf@+kJ
r!(x)=+ tf@-kl
(\ )
t-.-i,t 1++ u(x, ")= f(x) b; q ^+jl":(Y)
f*t-f ulrn " ) -- ' b; r.' b .,,.J":r*'o
ta;i ,rl-r
(r)
tr7 1+:r 1r ) jl 6r$lFl q
(f)
C*t-f (f ).1 (Y) g.+ ! .s"'l ..iU 4;:-tr k ;si g 6
t#,s qti (Y) il <r) ,rrt't'!
u(x,t)=+ [f@+ct)+k] + ] tfe-'r)-kl
u(x,t)= t / I [f(x+ct) +f(x-ct) ]
f(-ct)= -f(ct)
15 r-5 CU::-,| Jli g a uirj> )Jr y' ,Sbl q u(l,t) = " tSss )l tf(o+Yl)= f(a)
LS,l;l a'.f(x) on s r:-,1 t/ 0.2.1r tr rl *J^,lt.:*,1 or-i 1tL (0).lr Sf eU ,*-,t:L.f(x) t-f ')i*S;l; c*,-!"f .-*b ffigtl;tf rl,,l oJ-i .-c.,-yr; .5x{/
t--,C 1+l (\ ) rr gi ..nr5cr--1.dt- t
t-*'t l?,rJ u( o,t) = "
!-1
(0)
b; t n+;, tr.r
lJt;1..i
1l .:.-,1 cr-l\* (Y) d:--, .7lJ? ilr.rt 'f-u(x,t)- \/y [f @+ct)+f $-ct)] (?')
at;q+ s tui g'e,) +oliC i t'-st 6r.5.5r, b (f ) dL-' Vlr
u(\t)= 3 ancos rnuin+"i an= + [t.
&atie FL- u. Js -urls etr e!-l 6:r (v) : (?) =tr.r,
y'S ^'. +-S
.lrr5;,6{t+ tr gtr;.r .r3.+i .}til lt ?lf3r gr-l 0r.1,1
cos ),,,t sin \Lx=cos ry t*+r= { {srn{ ff t nr)l+sin{T g+ct)}
f*t-f (V).rr
u(lt): t/y,3, a,fin { + g-ct)I+ tr:i,o, skt + @+ct)I
i f-rf(x) tt't*Sf\l +,-sl tSf-r:t c5t+: x*ct yx-ct.lrt'i,Udt? U
f al= 3 a,fh +,ll= I
1,*tr..r t+i .i$t ,. i-,'t .r.l
u(rt)= + [f g+ct) + f @-ct1]
ey U &- U.ll *ilFl c,,..,,1: s Q) Vlrr S.5 r, ..1-,,1 (?) .l-r JLr +f
lt ?lf 'p t )tr L,et-t'tdr.ta f iJ &-.t f ;ols =lf e -*e3 Jlj ,l
,/r., y-l \ Jl:, ;frtl JLr 4L1-, nJ.ll *ilFl .+t a'y, .rr-, f.r r.b'.t y'.rl
c-1'-j3,!i -rt'y, ,* .,..,J b e c^:^iJ &-J f f ;tli g.li q qj q .Kti -r.ib
..:-,rl oJ-.i 1L:;lc= \ d:--, ,./-l.ll ..:.-rl oJ^i p-,,.2 ,= l&-J.P du&u,il= + tf@+ *, * f t - *,,,.JU+jf rn.rV \ Jg ,.>"^itq.oUiil&{ eof b:i-',u.tlf'#."ile.-f-
! ,?- ttt ,A?2> ^L4i | ^JtJ/ jl q I r I J(j J2t' u. e- ti'o tl.-l-j5 st
f(x)sinff* (v)
,)yati-:-^ !o)rL, ...(1. )
-;.: 3 r1i- &?2> +te; 1r ... )Yl 1/ .itJa-.J f J13.e f ss .n-i ,fJ) C \ ,.>.2y,a
..:-,1 glsxtt>,. J....tYlllsllr $A i J CJI +to;.;.r ...Jyl slls.,,Lp *iG-.j
*t1'r
,
utt'o', t:o
01
f ri'.|r
-,A-,0r
---"'.!''l'-!'' I r t=l/zc
:ffix.t)-2+,ix-r)=2'i='lc
lrr Jd
.)Ir-J-t .0.Y
-l {.ll t}.-,t t Pq3b )a, tS-tt;.: jl.r*l .t-rjJ- ,-r:Jt- t'b ,ib.ll +IL"'- \
.-rJU cr.i c.rl: ost' 1.li {rt .-ilFl
a. u(x,.)=x(\-x),ulx, o)= o . s-r< \
b. u(n ")=fsitla , uft o)= " o 33tt
u(x,t)--lcos ct sinx .vlr
t J)Cli )t ,-*- At JVrl drl -ilFl rfr1.SJ- Jl- rr b C ..;rbrl d:'-'- Y
)J-j +lf ./l;:tl ,f r5&'.! .r:? J)l-r ,ibtl e*it.JjL P il 4i uf
.(c=\) -r$g-.rr*.)b J-dl JeV el:t,SJt
6
u(\t)n=I
rkU=-a(n -a)
;F/\+(- \)n.'lcosnnct sin nlu
s-tn na cosnt sin nxnr
.:lr
.:lr€
n=l
u- Y A cosy t sky; - J cos?t sin?x+ =l-
co.r \ .t sin\ ..r- +...) .ylr+Nt 'f y? loo
,jy oli;* tl oYrlre (rv I
.&*rrl r.!r ! ,-l-* ,il.t nlt tj crYrL. jl 4, -f Vlf .y
,_1nc(x_9.:lr
u--rfr(xt+f ).Vlf
u=df ..itr?4=lnc(x+y|.-rtJ?
.:lr.JJJ- cJ-i olr 4;tr,=i:,. -# tf l.l -rj .tYrL, jl .5- r,r_ f
a.uo*un=lry ;v=xly , :=lx-lb. utluxy*uw=o i v=y, z -- x*lc.yun-nto*u, ;v=y , z=ry
otu , ortt l5J- ls li Jit'-' il &- .,,l- o
A.- f C'a= e I o<X<ft t)"Att arru(t.)Ecr'uttro)=. i c3xSn
u(.rt)=u(tt ,t)=uo(.,t)=c I uo(n,t)=sht i t2.
A. Uy*tlr= "
b. yur=ru,
c. rur=yuy
d un=v
e.uo*ux-1u= "
u=f(y)d +90)e','
u(tc,t) =EinfiE.xc_sinh{L n
c
O
,t= I
ir I sint
+Y,tc .YlJ?
.s*+6.r.,rJr /:
sin nt ct sin tu
b. us-\u*=x , o <x<rt i t) "
u(x, ")=1 , u{xr.) = \ *.r ,' . Sxstt
ur(",t)=t , ux@ rt)=1t i tZo
u(tt)- nt'-+? | +a;t+br+ ; {ancosYnt+b,sinYnt .vlr"' ln n=l
. # [FV"-\]lcosru. +x'+a
4.= \ -n/? , D.=Y , an= # [FV".'+\/ ,b,r= # ?V"
C. lla'Y04s=0 , o <.f < \ ,' t) "
u(x,.)= t* + , ut(x,.)= \ *r ,' o 3rS \
ur( ",t)= nll , ux(\ ,t)= " i t2 "
u(+t)=(o- * ", + . i'* + +. :=, t ffi cosdnnt
. "#
[FV"-t]sindnntl cosnnx+ +'- ?"
.+5 )>z=x-lt g v=x*\t ,riz .#! 3 r.rllr ,-)st t.lt J;Jiit -, jl &- i - ?
a. uu=tlts , o<x<fi , t)o
u(xr")=" ulxr.)=x* \ c3xSrt
U(",t)=" , U(ttrt)="
b. Ita- ltt' = x + tt c1x17t
u(xr")=\ , ur(\")=-x
u( "rt)=11 , u(rtrt) =t ;
tQ'ctlts = F(At) | o<-r</
u(X.)=f(x) i o sx<l
u( ",t) -p(t) i t> "
u(l,t)=q(t) i t2 o
l).
.3)c1tt
t2o
i t7o
Uf 4l.i*" .?.1
,r-tbA,l Jj l' ,,, J* q .,+ .rrl -l:
(M)
(6b)
(M)
(M)
Lf t-l c-,,1 )J, 1 -# il .-,13- J o)Jt .7l-l+ cSl-llr (0) d:.-, eS:b rlLS rlly.-r,
4Ll"'*, d/-l J- jl J+ Jr c*t* trr.':, 6jLu? ,tst q b ^l' .', ir-l :l-r+ .r5g,llr
ojf Jtist -)l s: 6l J* q dl.l.,. .2s r2lr> b.J) Jl-* d f*r.f dLr c.-;i
^I.n t*S..t ,f t ).b u-l cSlt JSat it'o (0) d:--, s> tJ-*r./ Jtt' d., -r)ry-{ oy.e +, J*J) W ojf JtaltJ -l-,t{ .:,.Lii f^ l+ 4^^ J) J o)1,-r<^"
ti J_p rlr-rr rI., ,*-t.i.,Lo I ,$ ;:.s1,.o ulslf JWt ii;t+ C!-, ;l 5
V je ;r.tx tF J* I lr.l.. ^+tt" .J-:U oJJ, Ce;-t" >t tStt-,1 -l: )l*g *- e J-P J ,Jt j -il .*b +La o.,lr> +-. J) .t<ji f-,.* I J-P +, 6leLa
tr rl., f t) JslF a>..;)J 4,. a)51 '>.2lrz l-..1) 4+1r".r;r -r.ol;'> u(x,t) J.tt'tr(OD) \\;,2>u ol(;i urt\q(t)t p(t) ,f(x) At4jtJ o),1 r.a-,:-;.r a.li--J-r,o
6l-o; drL, i ,f_(Oa) b;-;ru g.o: iLs 4<ri elt;-F $lrr jl-," (M)
( to ) lJ' J''i
o)U:-,| u)C irgr ! A-* r* +6lL-ql gAf rfljl ^5':"-'l ;j! trSuf
.rsl-
t J-pr ,gl.Li;;l.lL.,J'-l-l -;: 6r.,-L J i$'\ C-U-S .,lj 5-r-o.:./ .rL5 u.F
u i*r, 6|ao. P.:--L-., -".al.aVt S.,f t* *.,rls C!; P c^-L-,. U
. f*.rl: r-itr .lU J tvq +(;t+ it'u- c""'l Y""t=' &;'i dr/-l ;)r t !:*'
.0uo=-k -
s'dx
1.r o.rr5; -* oir.r p5rL;l b .l=,;lx+ M sx ,sV* t &'ei er tpt'q Jt'-
LQ, - LQ, o2;,e..li ;, 1---U: LQ, s LQ, 4i \lt rt'E rr d/-l;l Al ..lU; Jrt
U c,*,1 Al s o.ry Al it j JJt.rr & J-h ++L -rr o-t-! .jb ojf
LQ' LQ,:'t: rysAr+/< ry sN=k( ry ry)sAr
,r* ! 9..+ d ;r o.t-i .j! oi-r Jl$, d "r.orr' ib F,j +2r4s C-h :t
J-.t-ae'l y' 'b..lLj q \:*i oslf 4tJ) -# J) g..e
LQ, - LQr: nlrrt Nt=dP s A u L x
.C*t-"r 6t sV:L; +''?i l{
orW \u(x,t) rT' N=opNtLx
kop
kop
\ , \u(x+ Lx ,t) ilu(nt) ,
Lr' ax - a" I
t- -r
C*t-.f Ar+. Af'r. r(ii ,-il \0tu
-=AJct
+ = c'0rb;lj rr lofrf ,js 6Vh tJ.:,-l-re,*,-,; o \r.rt Ao lkop
('*t-tr0u , drtt
-:C'il Axl
ttliC b + 4,.rr rjf {+l) galJtl c,'.1F(at)=" gl;l q (0d) iJrl,^, JL^ r5
Oy.'e \ ,glf 6,t4:.^.+ q (O) d1,-, ,F Sll.r.llT r.a-,r q (O) d:.-, J-;tu(at) =v(tgt) +w(x,t)
,F.rr.t^+i.r.:Lr v(l,t)=o J v(.,t)=o 6 €,;.rt*;l ,Ss* lsw ,rtbtr-.,
€ rlf g,-,1 ./6C*e_t Cy d:-,u(r,t)=v(\t).ryx +p(t)
f),Jv L)lJ. tp i:"- q (0) b-l-;.lr cJrl: llj tt
Nt=-N
0u
at
VscrVr=f ,(Xr|
v(x,.)=f ,(x)
tt( ",t)= "
v(lt) = "
,' o <X<1,
I o-*S{
,
;
t)"t)"
t)"tlo
(ra1
Gu1
(rc1
(ra1
.li 1t rf
(rv) :^:.... .r'J..i
q rbL- d:.-, -1,:iLr dL, gg,Wlf\l.ri l.g_l u. * j L v dr#.r : (?).L-, S- Slt
,*jlr; J-S OS:e u. ,J.1tr LS,J4:-,.+ r, .:-,,1 .rltSC--,1 ey 4F..,"r 1t di
v(lt)= : ,Gn ft)sin +,
1; l:-,-J- .-rrtU .f Jtlells\tto=y i c <.r< \ ,' l) o
u(+ ") = \-costtx I o sr< \
u(o,t)=" i t2"
u( \,t) =l i t2 o
fJef :J-u(x,t) =v(x,t) +w(x,t)
€ rlf .:-,,1 JtlS ).&, U-l Sltv(\,t)=o J v(o,t)=o d('ot-"f Sttltw Iu(x,t) :v (x,t) * 1x t+,i ;l r w (x,t) = y,r
lj r-l \:-.,) t.V 6lrr..,;j +L.-. 6Jb C*l q .?i tl
vs\v'-=x
v(x, ")=\-costtx-lx , v(o,t)= o v(\,t)= "
.nt'd/$-q u
v(x,t)=":, Gn (t)sin run x
f (Gn +\n'n'Gn) sin ra*x= xC*t-..f r.2l r!--, drL,.2.1
i?, i'|'z' L^ \ &)rter ( f rf 1
-t-lt{r Cb r.p f*^, +'s-t, * l.lJ.Gn *\n1n'G" I €C ?U-:il St* ltG
,-*-G, *\n'n'Gn=y.[. ' oin nJunr
t+,i;t rGn *\n'n'G,,- Y (\)".' , n* o
wt
Gn =o,re --tor, + *; (-\)n*, .:'-'l J't 'f'u Yl'l+ tgbl: rJ'rL' yl
cr-lrtr@
v(4t)- t {ane -vtYzY' + + (\)"*'l sirurx\n'n
f*t-.f v(n ") = \ - cosltx -lxb; q .?i tl
f @n *=+= (-\)"*') sin nnx -- \ - cosnx - rx\n'ntl= I
t+l ;l-r
an=# (\)" a L t ry-I a l1$fi/. t ; n*t
o, = $ + y.[. '( r- ,orn* -lx) sin nx dx = +,:*lrh
u(x,t)- Yx- # e*". + siruu +
9b-J-t.V.Y
5s J.2lr> a+-l) 6 4ue-lr.t=rt=: lr,;.*;L.t J-rb q.l.r 4,ll Jif r'-..2.:- \
.(c' -- \.vOY) -t.lL * glijl .-& lll * t)21 '>tlr> r-. J) J o),. tt-n {, 2-a. u(x, ")- sin .. \ zr
u= sin o.\ fixe''vornrtlt" .:lr
.=lr
o. ,1-rtto=vl i ocr( \ , t7 o '"5J- l., Ji
'y. --il 4'"r -Y
u(x,")=sin-rrx i ogs\ ,
u(",1)=1 ;u(\,q=tt r t2o
b.usuo=1xtt i c<l(\ , t7"
u(x,.)=cos+ i csrs\ ,
u(.,t)=\, ux(l,t)=+ i t2.
| *; o<x<ob. f1x1- {
L \.-t ,' 0(x(\o
u-- + $in . | \n xe"rvorzrt -t /qsln o.fnxe '"' tvor(rz)rr
I x,' o<xcyoc. f(x)= {
L o"-t ,'Yo(x(o.
.,^Ui.U J,.b tf + '*-Ls,ly.e.lf Jta;l 4[.-, ;\.Y4,rr oslf 4>..;) cg$ o;ltJ (O) d:.-, elt '*i Jt! C{+, ,t , ":+;)
aa"La
drL, ;lx Jp +,61+Lii -tt tt &-J.lr rL,5-1r ;tslf a>.1) ir-l fQ..:,.-,1 rLb;4-,, 4r ;! r!:L, d/-l J- cslf g-+r $i ,J.li, :r {1.,[+ : - ,-l r.a-.,r t (Od )
Jr jf b !i t c.'., .-/-ltr 1 c-,,1 ,-tt l.t; J*t{ + r(!-l ! .bJ 1',Jl) L5lafl {11
(O) dr.-, i .rr5o.r[,i:..| rlly; (M) r (0c) b-lrJ,jl OeLrtf f]C./lr-ti i].tf C.,l,-r* yj d.?.*.r 4.r
It;crus ,' -o <.r( € (Vd )
u(t.)=f(x) ; -6 (x( o (VD)
(V) d:..., J- t{ 6c*,,1 ''i,*+.r:*,|
eLrz: ^J.ll
g-ltf a>.;) u(x,")=f(x)l.li;r ef
.:,-.,1 C.b dytr eslf JUtit d:-. .t3-l *,+t tr J.l J:*.r €.;l.r*.'t irli./orti:-,| lL--l 2t '>slr> Jti:it 'flf ;l :tli;6 d:.-, Jil J- .;) J.,-) f.^tf qJt+
d-r4 ,ilt r, d:.-, J- q )*, al ,slt *r,.r K- .;lr. q (V) 'd'". r Slt I \S
1'*t-..r Na).l,,.ri..r)t cr;&t* tr .gjlry,J u(x,t) =F (x) G (t)
GA_ ,D
c'Gt-.r
G-l<c'G=n , F-l<F=o
.r:,.-l t jl Ji:.,-, +S u(nt)=F(x) t+i;lr G(t)= \ f*U-./ .t<;T k=. 4..it ? Jb{i d/,hLi .-r-,tt crt;;l Jr:*. $li./ 4sl otv 4 t{ + .4 i ojf ae.lr
.,*t-.f oLf;i k=p't. n5,r. erJb r;,; Jrt-,Arft='. 6$l
tr=F(x)e"r" cr-brLr r G0=gctltrr
F-=F
nr*i t d t--J rl Stt.r!-r oif r+l) irt j.r'qlrl q.t..l,,l lift-:r f
d/-l q 1k=-wt." €net..,i)t{..:*,1 &,rtlL.7ll:.r.21;JLbt ,flf
r*t-.r ,-flG+c'n'G=. , F*w'F=o
r:.*r J: rt-re q .,3hlF rSl-rl: 4j r, rf
G (t) --s'c,w,t s F&) =a(w)coswx+b(w)sinwx
,itlJl+
u (4t,w ) = [ a (w ) coswx + b (w ) sinwx J s'ct wt t
,i#.t 4;l.y.r>-'l e;-rl.l w9b(w) :a(w) J-bt, r.., 4, ric..rl (Va) rJ.lt^. ,Sll, Alf
€ r"n).-/ .21'7 ;ta7 Y-rLir s,'-l
u@l: ! u$,t,w)dw
.r),SCrF-try tr.ll;r.f l-2 Lerl U-l e)r.o,-',1 (Va) rJrt^, SV,Jlf lu(x,t)
C*t-.f u(x, ")=f(x) b; +; * j tr .t.S-l'i (va ) drL, .p u(x,t)
@
u&.)= I [a@)coswx+b(w)sinwx] dw = f(x)
l+:ir ts .aiff(x) Cb.-lt JlFl v-l.r-;tljrSl2b(w) sa(w) *-l-r- d/-lJtf
@ *:*,,1 Jt Jlto \ ,llf rsbb (V) d:.-,
u@fl = t (a@)coswx+b(w)sinwx),'ctwl t4n
a(w) r -@ coswx( )=-r-J f@ ( ) dxb(w) lv -oo sinwx
,-rT;r +f
+ + ,rnrJ.b(n)= ll- .[llo
u(\t)= + I:
,)1;Oli*'Ug)rter (l.Y)
S ulua-lr l.rl* l-r.,fbb .I. Sr- 2s o2lr> a+t) .0 Jtlr
f x; l'lstu(x, ")= Jr"
L c t |tltr
ta(w)=. .',;l) ,ylY
xsinwxdx- Y I- x coswx
Jt' w
t Y ,sinw=-.-:-("''-'-cosw),tw w
\ ( sinw * cosw) sinwx , -ctwt t 4nww
ii f 5s J2lr> a>.;).5.-'rt - JrU tr.l.l 6-r: Ojf{>.J) JlrL.l:t'|-! rJL:
{>.Jr ,.b U-l Sll.:.r5orLi:.,11V) r!"-, Slt PV:lf 2l Jli;u.J,.11,i g../,) J)
\ 6 alr f#c -F A:^u ;* b- o ;13 .i-$x bl;l 't'f * ,stt oif
\- ,Sll ,-d-* Alf J dbti J:U tt \- Alt ,j*t C.lf t) e\:r,) !,f -f ,y-l
..:^-,1 I J* I; r;1.: .rlj *-k+ V P d.rti.rr drrL:. r.J .1... 6;) JjF JUrt d:-,
.1"# d.S.JHr: U.lly.f lt u(x,")=f(x),' o3(o Q5l '>slr- b.J) tsbl"
.)r-, ,trftrl )l ts:-p
fJt,-l t)tt cStu'{ t,1 .lt.- (r t t €C b.r t+.-l .rr b cgr^{ .5.,.}t-,
4 Jt-. ,SllS jt{L .b f-.-^t jl JrL q'JL C,ri" ylr+ -r.,_: frnlf aa,t ?
\f i,-t-t' -lt".jbljl Jr{ l, Jt-, dr::? {-r-)t b-lr-/ \:.-,.) er l-i.jbl .E-f+
.€g:VT s+e> 6Vi i ey4l.'.. "r J- q./U-ti ^I...5- i oslfJtirtd:--,
( l.r ).... .....lJrel..i
gtr-J*t .q.Y
,-51- A.t+l., -l: 4:t eU.S 6 ul:t';: -r;SJ- l.l ,f U:,.U .l.a ,.4 rr trf r!-.-- t
-t.i! y*.1 cl.;l;t
+5J- lt ti Jtl'-, jl 4, r'r- Y
| , ,'o(,u(Ya.f(x)= I ,f@)=f(x!
L " ,.r)Y
( v i o<-r<\I
b. f(x): J tr' ,' \ <.r< y f(-x)=-f(x)
L . ,' r)Y
I ,, i o1x1nA. Ut'lln= 1
L t ;x>rt
f w ,'o<-r<\u(x, ")= {
L -t ,'r) \
u( ",t) =t- \ | t2 "
. fv; lxl<nb.h-Yno= {L " ; lxl>n
u(x,.)={ l'l ; Itl<tt " ; lrl>t
rSJr{Jr dL.ai )t ey 4[..- ,y.\ o.Y
,'*tlte,:' ,'t 6blfr ^Jrl/-rU, t dL, ,F 3f- & ut+ i/_l.rr
ltr=cr(un+uil ,; c<.l1a, "<y<b2t7ou(x,y,.)=f(x,y) ; " Sxsa, . Sys,b
uF(r,J,.)=g&y) i o3x3a, .31<b
U=(Xr"rt)=c i .SXSA I l-.
u=(tcrbrt)=. i o3XSa, 12.
u=(.ryrt)=. ; o3l3brt>.
u=(aS4t)=" ;.3y3b,t2o
4F.".'r ul ,Sll ,Fi Jb .c"."1 lyy g;Jri 5> 6,Vi )5 Cy 4F. ."t l, d:--, gr-l
,J-re Ju r, rL, .;4l J- g\4.c'-,lb 1a cst{Jj t d!d*. .t} 6- ,ibllOlfe r, dl, ,SV, r-elf 6t4;..,+ {., C-;i ,€r.l F
u(\y,t)=F(tilG(t)
FG=c'(Fo+Frr)G
*= + (F,,,Fy)=kt
f),1-$ c,,)rL,. q t4.rT jl1 c*,1 ,.Lrti 6;l-tr k, tsi J) +5
Fs+Fyy-krF=o, G-k, ctG=.
\ ,Af 6,t4t-..+ q s.."1 jf i:j, b +JrL, 4. y )Jt 6J9l rl.rL, ,F All
C*t-.r drL,1r .ri d&V tr .gjby cF(x,y)=H(x)Q 0);'trQI{ +HQ'-ktHQ="
,)1.r.ol;:*!Olrtre .(l.f)
(t.u)
(^b)
(t'r)
(td)
(te)
(^f)
(t s)
C*t-.f 0@ * ii Ctt 6.rs lr .r'-tbe,-/
f*rlr c'FG J cgs1-j U-l ii-h 1'r-it t{
b-1
+ = [ tr,o-Q'):k'
F)€ at );t,r;-.r crirt^, q l+"i;1.,
I{ -krH= "
Q' '&rk,)Q='
Q : H st-e y: \tr !u(x,y,t)=H(x)Q0)G(t) .r (AS) tt (M) \t-; 6-tt il
€),/Q(,):. , H(a)=" , H(o)="
d:*, f{Jr-, J;t ,:; ,, q;i .r.:;L.r
H( "1= " H(a)= "
Q@)=. ,
Ir' -k,H= "
.:-,l.i,!o+L..,, ,l-l=lrskr=-ff,^=o, t \, tYr.-. r5.:*"1 =lr,JlJl) ePs
4,1?.*,, )*'--6 s H,n@)=sin ff, :,
Q" -(krkr)Q=o , Q(o)=" , Q@)="
6,114.,>-,lQr|) :sin ffl ?lr 6,litk,-kr=- #- ; n= o, t \, tY, ..- csbl q
S €rf +i c*;'.; .7-.tl>t,r-/t3-k,c'G= o +J:Lr J* q G ir-f oat'-,-k,:-k,,. + : .t' ( # * # I
,-?t\)rrrur:Cxt'/, # * # IGr*r(t) = o,rurcoslt,rurt +b,*Sin)rrrul,5tj*
=lf cSblr 6G*)r'rr^ C= o r,;lr
t.p-l )l r:*.,1
urr*(W,t)=(arurcos).r*rt +b,ru;in1,,urt) sin ry, tU ff I
,)2Oti;*!o)rt'e .(t.r)
a € '
fln .h :t JJ J)-t'1 (A) d:'-' sll. ,J.lr JJ:SI eu(4y,t)
a bY
u(x,y,.) = i, :,a^,otin ry,,* f y = f(x,y)
. ,-P-'*Jf cb .rlt '" ?r4 blic l)or* d/lJt{Y fo b
arnn= i ! lffwt'*ff. 'ufft dytu
@€
ur(xryr ") = In=l n= I
.mfiwtsux-x t*Tl = g@,y)
C*t-.-r (tc) s9'5t tl
ta"l ;lr
b,,*= # {" I,ortr,rlsinYx ,n ffl dyrt
qJ.lf ,ik.rl t -f*.J-ll u.-,n a--b=n ,c= \ .5ljl q l, e, +l:,..., .? Jlbtb^n= " f*U-".f ,lgl
'>,cr-, )-ttJ ;re cJ.e t .l$J- f&y) = ry sinx siny
'i
.'f ,in m x sinny dr dy: * ! , siru sin mx dx I irny sin ny dy
\^((:\)'"*='-\) ; m*\(m'-\)'
anvt= fi I.
[.;iro sin mx fla =
{ trt'
f ; m=\
' . fJr Jl"ai
f*t-,r
\ t mn((-ll"_+'-)) (-l)'-'+'-l) ; m*\;n r \fii 1m' -\ )' (n'- \ )
t
nm((-\)'"+-'-\) ; m*\,.tr= \ ; )rur,= ,/(m, +nr)
(m'-\)'orut =
+ ; m=\, z=\
u:n,rrl,r ^;t+ .r:-,r-l JeV4J.l--. :lf u(x,l,t) .obnvr t a,,^ JsfCf;it- U
4;lr2 oJt^i.jbl J- .b jl Jli C JtV q b t5rr tt ey d:"- :lr f*t-SorrLl
Jilu s ,S4l>Uf i:-,J- q ttli r.l.li rsLl 'ttsi r'l'-'rr q d.l*t &-ey 4ll"-
,pl:',r .r)t J cr).:L-, q b .-rrj d-^' d/-lJtl ti.il':y dl''t u Cy '' UJ
F lt tgi q bly Jilu-, ,sl-f 6tL:.: tr J,-r f."tf ^+tt" 'f+trf ,:'ol-a:*l
.le-r J- Jrt t grlc;rri S ")W:t
u. *i tr Ulv.f
sqJ*t .\ \ 'Y
Q5l ,-zor-,l5+k et*l: c=\ J a=b= \ J<.i el ob Jq iUlrl- \
j* J.lJ.qtrl ilFl.r $t: ,ibtl ,-rr* .Ifu Cr, lt-ti dtt{ i o>2'sit'e rlib.ll.uitr y*; Clj;t
a) f(Xfl=sintrx sinlny ; u = cosn {6 *itt^trxsintry
b) f(nfl = o I o \xy (t-x)(\-y)
u= + ,:, ,i,# cos(nt'l(*'+n')) sinmnxsinntry
.vl-r
vlr
.' '-a tf "rrJ rl-t-el fl Jftt.li-;r..5
,rlgJ d\) .r)rL' ,tr.\ Y.Y
Ui : ,J-f Cryly jl .*_,i +, nf lt-.| t ,-*rSJ ,Stot! ;-,, r^ilr J q)l+ SVsi.r>tJ d.rt^, :lr I >r fq Cb ii t r*L.Lr',fb, gb LSbb rS6.6sEe5-ui! s:+y Sf S:,v Ctt
.t+t? F ,Sjtt Cbr" csl;lq d,*Sf .st-rirL_.c-.,1
..:-,1 F ,SjtJ.Clr.-ttlr", Ctr/c5)Jr Clr f="f .t<ji Yf=F
;lr d-aU t ^S-u! €' t€.-&Fttti-lr drd t)t+ ,Ss-,:F: kee' | -+q L
JL:r 6^F = -kee' V ( + ) .i;4-*l ilygulrJlj.t<,i -r,rb ,tS -S-e-
Cli .c^-,1 ,.r\Y +J.:L, =lr + .I+1", Cb -r .:-,1 j-tq 6b LSbl: F -r.o;rr,
.,,r.,nb ;;7 bb.,/;;,-r<"U-,1:-;SJljhttt, l, ,5'-rrgl gto-r; jl e j*+Ir",
,5-.;r .r\Y rJrL, J- q d/_lJLl ..:-,1 "r\) d:t^, ?lf .:-;l-ri )frsrl* drtLi;):+y e siJhi, ;ti ;l gS j.rU-rtj! t €)e/ dl+. ,.ri;, ):.-f,j"rt ,, t .rl+.
J qrt? Sbsi Lgt+l+,. J'tt -, 2s ;121K y.oi)te .r\i aJ:L, .C*t_.-f b .11.r.,'. ,.li -,,,
+f11t, c.Jt- ss ,.> jr> JU:;t Ji3t -, J- -1r 41-? :1 ,5.&_t Lp LSLnryJKcSbb uil.r
J$t---. -;l .f)t*, .lt*-f ,_r-p I c^-"; c*,11 .JL-j ! Oslf {>. J) .f lr;;. 1-tbar./ JU, -t t -5, q -lL, dri Fst 6,ly,.>*,1 .r<.rQl.lt$
64-P t{ }U, ,2tx 22>,t -,1)-t' E 6tl-rd)t^ .,^ieJr dd,JU:,..,1.-; q.V J[1,
:it'i +, .i[..n.a ,l-l tSss ),r+-f .Jt",Ftt, .iy:il 6 ,Jry.,rr {:L: .!:-^ 0 -l- 0
.-tl-lL cJ-l Yl. ; \ \.
S -u)r-; J)un=o J23,.e q,.f4! rJ.:Lr.fq-" ,>Vie )37 scJ[r r,, .ry
. . 3.rr ;[.ni
,*t-f u(-6)- \ \ o , u(6)= YY . b-V !+i !..:*,,1 u(x)-sx*b .:1'5+ 6bl)
.:-,1.F: \ \i J.lJdu=.,FFt 6ti u(x)- \ w+ \ f0,thV. sb= \ 20 3a= \ \
..:-"1 )t.'9,j"rttt-" '>Vi-.a J 6
,J)t' 1:.*k b \' r0,SQpL; qr.s-r*fo dl-l-^'llr,.,r:: t:+-ffb' A JU"
.!XU cJ.l YY. I \ \. .r:ti r., br;1..:-,1 ,l-l -t:try,JbFq 'ii':tl d
ttn* | "r* \ un+uo= o J23,e e, 6;lcl,.:-'l 'iL.a:;u lt ,-f\! drL' '
J-;f l, -b ttf "l..;t",.:*,1 ',ilsfkt-. 4;bcl;-,1 tr-, ! +i L 5c*''l
f;Si \:,-,,) u. -/-S 4[--,
ur* \ ,r-o ; u(0)=\\o, u(\.)=YY"r
)r.'., 6fit.,fut\su,= l|rt"T =tn) ."u-r +, = - * ,'.e-nlFltt
f-rlru=oLnr*b
f-tsi rrgr') +'. Gi-t' \l-;q qi t* r
u:#Lnr*\\.(r-#,or;1 U-!c.Jl,-.,1) \o -10 cSt6-otj *ofs, U,d:l-at 2;.>ir- 3-t{+t ".q
JL:,
of S, Jtl, 2>12,.>2lr- aa.l) CtJ -tJ!0 -l \ " '.}t'i :tsl-f 4>.J) .,l:'r*j { Lno-S-l
.r=tk
Jy.e +t 6Sf .iLa:;r .1) .r"S) d:t^'
!e aY 1* \
0r' 0r' sinQ
a0u\0'u$ (smP
nO )* ,UV T ="
oljCls o-S s> U. LSI-at .tt Jsl-r 4+-rl Jl$, L-r-.o csLn'-S )trs ..:J-" o +5':"'l
jS OW'^ ! O)r1....
r.;1i .:*,r t.lj aJl-"., J-- ;t
* r, # r=. , u(5)=1" , u(\./=o
C*t-,-f +Il"-"', eJrL, q qi q
u=- L +b L. ,' ou -orJ1rt-f r-1L".+ ,\F".", 6ltlf b-lrJ
-,11 o.:l..c-,| b
,_ 6o
rd/-l J- -<;* ",",jb;.,f ctrn p 4iV) f g-,ly t-.r)t) 4L,.,, J"- q i15l e
€C *u d:.-r.5-- JP ! tr,-Ft*, "l
y-; "lrL ,!-'
c*'l '-,-)L..\' Jt'1,
uft+u*--x+ly i o<x<fi t o<y<xt
u(xr")=x i o=x*Jt
u(x,n)-1 | oSx<n
u( ,,y) =y ,' o Sy=lt
u(n,y)=cory i o.y.n
tv(x,o)=o d.ot*"f iLi" ltw s u(n|:v(x,y)+w(x,y) C=ar"f ,lf .h
J dt-. b-lrJ t 4+j \ sw(x,y)-ay*b €,y otl )-,b- u-l ,Sltv(x,n)=.,
f:l: r+;i;l ,o= # , b--x C*t-f iy .sLn;t=,
u(x,y)=v(x,y) a | ! y+x, w(x,y) - $ r*,C*t-f :1ai t'.u(n,y)-coV su(',y)=y !.?i t:
v(n,y)=coV- #r-" t v(.,y)=(\- * r,f--rlr
v 6lt\ ti d:^-" d/lJtlvs * vr--x*ly ; v(4 o)= o
... fJ! ;l''i
v(x,n)=o , v(o,l)= O-! )l , v(n,y)=coV- #r-"
C*t-f vn * vw=. eJ.:L,.2, .li et t v(x,y)=F(x) G(y) et\F-Gn-LFG-
. f )er, il J;tAl 'J>L,lr t l+'l jl.l
Gn *kG --. t F'-kF= "
Wi;f sG@) =" t G(-)--. e*t-f v \blld:-.-lJ)JJ-rt hly'lqitlGr(y)= sinnl 1 k=n' r,*rlr
@ Jtt' u.d.ly 6,t=:.*-T 1v q b-y.y d:"- 1- eUiJSl f"
v(x,y)--
f*t-.-f J:L,;: rJi 6t: \ '1sl>*/
2lo"tn'Fu)sin ny =x+Y)
F,, -n,Fn = + I h*g)sin ny dy= a,p*b,,
6rot-r c:-hrrr' [+i;lr
oi -2r +f
att= *[(V"-\l ,b,= ]c\)'*';l .:,*.,1 .r:!t iY rJrL'
'/-f ?l-f S
F,,(x)= -#** +A,do +8,,e-'8 '-l+l jlr
v(+y)= i r- 7*fr +e,eo +n,,e-'o)sinnY
C*t-r qi a', v a.l:'-, + b-r.-t ptra t lr \ltJ q qi ti
jyol;;*!oIrt, ..(r\y)
6
v(",f)=n=I
,- bnI, n'
*.|o-* ) I
*An*B,,Jsin ny = f V| )l
y sin ny dy
V(n,y)= i f * - 4 o +An{n +Bne'MJsin ny=cosy -l-x-- lt-Jtxt
t+i,l-rAn+Bn-
Ar{o+Bne'/t1t- *.4""n
tror! + y-.t) sin ny dy+L I
L. S t-*i,t dE\-rv(x,y) .li q .tj !, C=f.f r-,1'.r ltBn ln eJrL, lr .*l J- t{
)-*e JeV -h ,r-rd:--, :lr u(+y)-v(x,y)a I r y+x-;r.ri 6-r:
4,Slt C-f l, ,F : ,Stf.ita:;- tr .r*) d:-, J- q o-r:,_i ,l;r -;r
J** .r r1.,,i9 cr).rL, CVlf c;L-l q jt, ./3t -, d,:i? S- Sll.g,jlrl.''. oylr
u jl gt^;l+ 6jt?rilr t#i oolr :r it ,/)J.r kr:e o,,r dbr t "1,-i
-2: .g,--;r:
J-rrrllj.rY.:L, C\lr tr,*:.: j9i dlJ rrl!-i){r.g.,jlr J,,-/ J5ijbJ.l)Js;e t +lrL. rr.-t:5 u-lly J.;t .-, .r)rL, :5q -uJj r;.
(\-x'1G'-lxG' + n(n*\) G =.rJrJrL, cr-l.shl-r+ jl ,-&..:-,1 lyy-,r$$ +lrL, rr.:-,,1 * 6;)Jen.li-,1: +5
! .:.-"t tl-t:.1-,1 lf y.l$$ Ltl&? J:" q
p,(x)=ffi*'- ffir^ +...=
(\\r) ' lrr Jl''i
Mt G\)," -. .(.tn-tm)t {-,^t =o Y"m!(n-m)!(n-lm)! -
.rT q .+.r, q .-r-,q )-,l t- e)n 6 ; s.r^di ! M= | fr- \) L_M= t ai r.r r5
e*t-trp.(x)-\ , p,@)=x , p.,@)= + (\"'-\) ,p,@)= * ,*'-yx) ,
pr(x) = | ft*'-t,.r'+y;, po@)= * Oyxo-v.xr+ 1 6rr)
: g,rlr 3 -t;J-.L;.. -\ S.r< \ d.rl,i ,t {Pn$)}r,=, .rsg Cl;.JLi, r5+.5 +i
., | " ;n*mJ p"&)p",(x)dx- 1 ,; n=mt Yn*\
-r+ jf .-Pt ..b 6r ,S-- '>,2j"," !, ilj r./ ltl=f$); I - \ Sx< \ ,>25,.o t Cb ro
@ ,.1:lr;lj ! i'r -2r r:._:; -rs$ a;tr6hL-f(9-2 o, pn@)
C*t-r \ tt - t r,Lati .1, .ri :l s$|Ssrl sp*@) y: dT d.h ?-re s
om: tt* '' I '[g)p,ng)dxY
arr=9 t'f@)p,,(x)dxY
gll: n u.m j2l* L. 5
,);r'oti:.L!di)rtl (t\f)
Oy.o o rJrLr -p .gjl:y ,-/ ,h.JrL, ,f)J. { IJJ.5I f.lxtw' *xw' + (i'-vt )tA) = c
,Jt 4 ,Sl;l q ..:-,,1 lyy {, q.,rjl J-t rJ:L, r, .:*,1 ,r+ .5)-te , ai tr d
.:..,,1 JJ O1p I ,-F ,Yu Vl-l+ 1r 4i!lr d:t^, dr.l v y *J-r(x)=v-v ; ,== rr('\|lx"n
lfn'v m! f (m-v+ \/
@
Jr(x1=vvffl= o
(-\)^ x'^YT
tn+Y m! T(v +m+ \)
,*;) rll lyy t,6gti + T (a)=
I (a+\) = af (a)
f;lr o,=n f )J, y' 6,1;l u.5
T (n* \) =n!
t o)t ,V c-'ll Jur&) sJn@) =lf jr o63i v=n -t-,t g--a ts;)J, v q.jL+
f{rltJ-"(x)=(-\)" J,,(x)
-ul.irLrt4iy-rLL sJoj>.s-iJ.6lolJc.-..-.rl;l .r:,jL5tci^, iJ,(x) s J.(x) Clj;l
l@=\-Tfu-# #.
@
! e't f ' dt.li .2.: r'
J.hI_ x - x' + ro _rr' I Y'(l!)(l!) tofi!)(Y!)
x"
r'ffr)(t0+ ...
&iJ .1 -F t.:-,,1 ,--r;KJ d-? dr- a;l-;lr J-r d:t^, g-l: yl;+ v=n.:Jt- ;.: ;
..]-,,1 JT O,'3*
-ul.i-;Le ii P jl .S*Jjl ,-rl;- C!, J o.r;, .-,19; -:"-!:.1 6l.sl:J.(x) = " dtLr
jl
a,=y/f .11, , ar=6/0y. \ , dr=Alt0fv , er=\ \/vl\0 , e6=\f /qf "f, li.Jt,r .t<ji t)L.I.(x)=o r.JrLr cSklf ;l ol;Jr.JFt,51 tar,)n=, .t#t
"(Jr
,r.*r- JuJrLrx ,.)jJ Ctt {.t g-jR c.rtj )-re y' U P drU 2t U.(arfi)}n=,
,>-:,r, ?.p F.t F Cl" )l er &.-t JliC\y=f(x) 1 . (r(R Cbr"6&it- e t* llan '4V *)-* ,?-
flx) -@
ll= c
anl.(a,p)
n f. ;n*m[.rJ.(o,,x)J.(a,,,x)dx=
1 R, "f , @n) ; n=mtY
I.t-* rS r-,L, an ,.-t-;^b iy glL; )t ffi dt? q n Jtf ;,r-fit+ tf .l
of 4,St r.1.\I4U.-, ,y.\f.YGlt s plrl;.R LE u. of
'S'- S|;'.r>tJ d:..- l- J"f= e', u:+ .l-l t.:.r-'l! 0 r,-.rlj jl Jft^-, t .)-t2 Lla:^-, of 6u,,f..rq Jh.A, €g ,-f "f ):br 0ll
elre o" d/-l q +i tt.-r..,t{ Q :l ,f.V oye !,f P J "[r;tt, p-
/-1 rjr.*. J- q .:,-.,1 ,Sg t.b U-l Sll.6k ,f .*1V.l .[-lr JrtO;r l, ,]rJh
t*jll.nfY'tt=" , u(&",Q)=f@)
p*tr.r R V P &U 9 6r,Atfrt sxJ.(a,,{).2.r .ri d-h ?-t' ! gle.rr
Yu2n-M x f(x) I,(a,,,x)dx
,)yol;zUle)rtl ..(\\t)
4 C'r.2r 6kL--, ffi..:*,| ,.5 4-'l .J't.- ?lf J et-P ,frUf(il .li.2r r5
.t-1,-1'1-,,: 1r.rfJtl, i )-t f ,Jt.6.!-iln, t{; Ji J- ti e-*,1 ,rbl:4lj-'*,
f*rlr Js otre q lU b; t jt+ ,f CtV i )-t t.-Ltq d)-S rat"-' 6,1J.
lim u(r,O,Q)= "f+a
-;: J.;h of gV i )t f ,Jtd.Jl",.r)-51{r 6)V d:.-, ,F Slt e-.1-tltl€ d:.-, ,y -<f= u. '.>)t'a y' -lr t{.21:-':.-l );r JIJ +q tl'*'-k+
r:-,1 J-s Ol-te 4. 6J-5.iL.ci, Jr ,-,r)L,! .JrL, S €,; +i o*;J
\ o', _osin'f A0'
f-.rl: 0 *u ;l*,.tt;-rr cJr r, r!-.- ,J-l i s
u(r,O)= F(r)G(Q) ,,>1,-o \ ,Alf ,S# t ^F." , ,F ell
F d {.*trr iY drt^' -,'t ..li d)-5.i1-&t?
)+ rinO *6inQffit="t_ _r
' \ d ,-r-^'dG I,= Ltsin.h-l-li' - csint aO rusrrz dO '
gtrl I J;;l*-: rJ>L, -rr t t+;i.llr
fiefft.# frouo#,*
*cfft.# frr'uofft="
Ll Sjlr,',rf
Gd 6,dldr' dr
t d ., dF:- I|',-
F drt' dr
*,r# )-kF=.
#*ftv*offit *kG="
(\\V).... ....lrtFi
.,i,-r+ -,sH +lrL^. q lJ, rJrL, w -cosQ ,r.t,J"r'.t q ..:*,1 ,t-ll rJrt-', .5- Jrt rJ:L,
*. e-y ) el Cr-l q + j ! C'-, -tt t-..-- r./
sin'O= \-wY, # = * # --'i"A *i),-l f J) ot-rp + iy rJrL,
#,,\-w') ffi t*kG:.e, rJ.:L,.t<ji.:-,1 * LS)bn ii 2> rIk=n(n+ \) f$.,rk;l ffi ^+t "
Jb
Jtt'
( \-w' )G" ,r-lwG', n*n(n* \ )Gn: "
Gn@):Pn@) f-#rS.,sg gln +-,2> 6ld.? 4 ,l1l.sG,,(w) ..x-i u. s:
€),-/.tr-rl 'drL, u.F \ bxt 'JrL' .2tk=n(n+ \) Qt lr.i i,s,l er'F'+lrF'-n(n* \)F= "
otp \'l*':lf 4;Ll) drL,;r-l
F,r(r) : otr r" + h@
I c.*,1 ttV d:^-- :lf I c-,'l
u(r,Q )- i (an r"* .L ) p,, (cosQ)
ell ,t-llq -t^iq .:,-tg -rulyr-, .,I5.JL- l rf f -t-r, ,jr-b, rS;i c.47 u.Js
t+i rt Jbn=o € ,"f g*,1 p) .t'rl-lr JaUt
u(r,Q ) - :. o,, r" p,, (cosQ)
@ c*t-r u(R'f )= f@) b; q '?i tl
C*tr.r Q =cos- 'w a-,nr :lt f@);r {+,t-a
jh AW'^ U.i)rt r ....( t lA )
f@)-- f (cos-'w)=g(w)
t+"T;l.r€
;, dbhL? !a -J
Gb..r-l L-{.1r s@)gb.*,; 1,"11lta,,Pd/lJtl
C*t-rf .ri q +i\: p$,1ln+\'an=ft ! g@)p,(w)dw
fe;T;f Jan=. frfr..t-, ,t-a rz.-lA;r J:;L, r5-i c.g q ofctB q-b cgh
u(r,Q )= ,:. h pn@osQ)
@ \eu(&Q )= f@) b; t qi t{
u(&Q)= t + pn@osQ)= f@ )
J,' ,'or ,;..7 l2bn,>,-,t jtt.u *if ors,ptr ,sll Jt"r, ^r,i -u;Lr
bn =-Y?-.P+' I'grw)pn@)dw,.Y
y.tu4..>u .(-,jbrl d:..- b.\ f .t'ulf c;u JVssl €e et t glC .h t trR Lu q Jg 6;loyl: cslr=-b
.rr u,ib.rlJ .l:-iL eft-p oi;W,l ^Jrl
u2-,.r d.rl ,ib.rl ro)t -F ;2llreb
.)J'"r; .lt* y-,1 oy,e + ttlig l, 6ld-, ffi.r"Sr=l uevl t* u. k, .=-b
;l .:...1 Olt r p c,rt^a:;., J: cShL--, ffi rJrL,
tta=ct (trr* + ur+ ) us)
(\\t ).... ...l.rtel'.i
c*-i r:-,1; g .., .li \iL#iJ J ,)f ,"rJ ueVS )* \ r1lb .,l) .-)ib-.rl {<!-l +. }r:
t) ,-/ ), J) rs2e + jt rJrL,
uu=c,(yn+ L u)r
f-rlr +Jl..., \1;.31.+i ti
u(r,")=f(r) , Lttt,.)=gt) , u(Rt)="
C*t-f rJ:L, .2r r.ri d)-S cr--idb 5 u(r,t) =w(r)G (t) ,-lt U
+= \ (w'+ In') = -k' U-1 wG=c' (w'+ Lw') Gc' s w ' r ' -'r ' r '
d:.-, d,t;.Jl dr:i? tr +t f="tf {+tt" J g,,-lolf tt-;l P b -U rrr l5:r-l .2r
f"llr iy cr)rL, r, +i tr.tf $l.1> :lf cSl.rl:
G+c'KG=. , w'* j w' *k'w=or
.t-'j J* pfu 1il.f*, +J:t r, rJrL, b f+r.f s=l<r r-e-#;.lr eJrL, tr
f-,rb, dw dwds dw
-
tD
dr ds dr ds
)jb4't tw'=k' d'w
ds' - ,-l s) Js oupe, rJlL, t+i;l-l
d'w \ dw
-+ :+w:ods' .r ds
d/-lJtf .c-,,|,I.(s ) ii ,-,1- s l-t\C ?.k *.-rjl J-.t drL, .-(,- 5w(r) =J.1s) =J.(l<r)
r*L r u(&t)- " b; q qi q
I.(kR):. \ew(R)-"
,--r- -ulb "I. .gtrr-:r-, tltffi 6fo$Vl;^sl 6;1jb !& 5c,.-,,1 p) t+T;l.l
C*t-r .li q +i b -r::"..e J. 6ll.u:utl.n'n ;lt = \ , Y,.. .)i.l' k k= ?
f*t-rf G Yb*tdrL^, ))k= ? .'lt\dn+),n' Gn= o i An=, ?
d/lJt+.c-,,1 G"(t)=o,rcoslnt + boshAnt ?1fr a;|.;lr 6
u(r,t )= (ancoslnt + b,sintnt) J. ( 7 ,l
wnt)=J.( fr ,l
€
n=o
@
n=ou(r,. )= anJ. (? rl = ft)
€C ortq",,l d:--, ^J;l
Jr-|y' jlbn san ;r*,s 6|111
e-$g..li q .+J, I'i -t
r-r-*A J
Yun - RV 1"")I R
rf(r)J.
uj(rr. ):
bn = to, , I* ,g(r)t.CdnRIt'@il o
S u-t-t' s)t1.t'.i.(J ^'.
jF gLi:.lr q,JiL- ,y.\0.Y.f\).r\l+
.2.r .cil: a jf .iti:.:, tr Jt-, J- q JIiCI .r\Y s {:-f .i\-r"; &f r
&+ e,t-.rr.; o g:il:; jl J+ t-.:,-l 1: J f*$ t-,,i ,r,ob,.b .*lt cr\ry ! J-ll }*tc.,L&.:, q ,jtt-r;l rL..- J+ J- t .i\.r; ,t-l \y 2> $ r-a -l .raLr +_-t!
(froar
(%rIdr'R'C*t-.f .ll q .+;, ! -l
'"'' lJr !t"ti
U .i).rL, ,)- 6f* JU,5--.jbl !.lt-!t) : gsltXd.i$--rn;al .S'Jo if.r."rF :lt ,-rst ttf lt./)t) .-l'-'* 5'J u. ir c',Lc':'
[.,n] d.aU J L ,,ob. +Lrti ..(r- .;r 4;h* u7^t ,f.V f(x) €,; rt-f
,*J f.rr./ ,:"-L;fr(f) r, 5 bf&) Cb.-*t - +-tl d-r1--' "1"-"''Kii .J-lb
dc'-'u't
F'(f) = f(x)sin nx dx = fr(n) ; n: \, Y,f... :
€C -L-y; & l\ c-fu,r,oL:. u-sf d*..l.,_r",
f(x) - ,:, f,(n)sin ra
€,y,-L-y; ffi q j u.l2 uriS.'.,r,otr +-s! J,Pt - u-s! ,-/# cr\-ry
F,{f}: f(x)cos nx dx = fr(n) ; n= o, \, Y,...
Y ["Jto
Y ["lvo
f(x)- | f,{")* > f,@)cos nx
c.iu) dr-5.-.,- j;r
.Kii rru I .,nl
F,V,l: + ff(")-(-tl' iln) l-n'F,{fI
F,V'I- Y [(t)"f (n)-f (")l-n'F,{ftxt
{Lrti t sl6s ^:-'raf @) :f (x) 06y'
(-'5
.+5 ,F utt.l,-+,5.5 +,.1t J14l:--, .\ Jtl,It11-c\uo=1 ' o<r<fi , t)o
,);7 ot;:,-, ! 9)rtre
u(xr")-" i ulqo):o , o=x<1t
u( ",t)= " ; u(nrt)=. , 12 o
"1.,-r+ t .65f Jp \ ..rU Cb tSt+ q G(+t)
'p- F c.Jt- -r)l-r 4L',",,, .J-,:-,d 1++ rJ:L, ll elu:y
Fr{uol - ct Fr{uo)' -Fr{G (r,t)I
F,{uu}= + I.n u,,rin nx dx = # ,* I "r6,t)sin nr cbl : d'v(n't)dtl
f-.llr C.h -,11 .;-,1 u(x,t) .-4b-. ,-/p
"Jr,-r", v(n,t),.1i -;.: 6
Fr{uol= + [u(.,r)-(-\)" u(n,t)]-n'\ v(n,t) = -n'v(n,t)
C*t-.f .t<ii t;\G(x,t) ,?l- dp "1,-r",
Fr(n,t) ^+l'+
#. n'c'v=Fr(n,t)=F,{t}= } t\+(- \)n*'l
y-2 J2;o u. ,Jlf LSblr 6,:*,1 7s> Ft ,V "L*,t:*-l
drL, .-i- +lrL, ;r_l
r:-rl
v(n,t) = Acos n ct + B sin nct i #/t + (-\),'*,1
F,{u(x,.)}= I f n
u(r,")sin ru dx = v(^,.r=ft-t li L'-l; d,v ';6rr u
JT
t
t B:. f"rlt
( I Yt ) ... . lJf J''i
v(n,t)= #/\+(-\)"*'/(\ - cosnct)
u(x|= + v(.,.) *,:,
Yv(n,")= # [(-t)"-t] ;
tnt )l
Ort rtf
y; d:.-, J- *l :-)b.Y Jtb
nfi,T,*+!.'
u(x,0= * :, ry (\ - cosnct) sin ru
ur--uo*u
u(x, ") =f(x)
; .1X11 t)o,
tty(o,t)=o ur(l,t)="
ol eJrLr U-i"h :l e-S ,fr;JJ'-'r- l.'r''iu.u(nt) Lt-tt""S.J*'*V(n't) *+1""
p*t-.r 6,lr,lf \l--J, dr;115
v, * (n'-\) v=.
;l c--,,1 crlt , 6 :lf I
V(n t)-sr'(ttl't)t
fle {+r.i drt hlr clr;1t5t b
V (n,t) =V (n, . ) s' (nt' t)t
.li 2r +5
V(n, ")= f(x)cos
;l .- *'l s.rt t :lf Or.lrrtf
V(n,t)cos + ,
tV(',o) =tt g.,;ll f(x)=x;"<scltt tsljl q
2= \r Yrf,...
;l .r,,1 ,>1Vf(x):x A;lil + d:.-, :lf I
,jy ati;* ! ri).rtl.
u(x,t)-- +.: , * t(tf-\J s-@'\'i'cos nx
6 ,J.ry.lr J::t* l.1 ,rU:,U d., ,4 f Jj,r> {+-r) .f JU,u(x,
" ) =f(x) ; -€ <r< €
f;f *e 6,?sur(4t)+o , u(x,t)+. -'lrl>"9'sl;l+;5
C*t-f drL, tt ,f -S ,tL:.b r,__;y .J*r+ t, .Jl-F{u,} :c'F{uol =c' (-w')F{uI =-c'w'tt
f__rl: ,j-b;l ..:*,1 u qug &r+ u ii 2t 6
F{u,I= # !_* u, r'i*dx = # * [** ur-i*ax: a#
d/-lJtldu
- = -c'w't
at
Vl-r+ cSblr J3l +sr'.J=*,t +,.: rJrL,;,rl
tt(w,t) =a(w) ,'ctwtt
d/lJLt stt(w,o) : a(w) :7@) : Ft\fl 4:l Lr cgu )1..:,-,ttt(w,t) -- 7@) ,-ctwtt
f--rl: 6$ ntS,^' "t--r+
t:
u(x,t)= # I *' flr) ,-ctwtt ei* cl*
it*l-- # I_**ftD e'i* d,
d)-5cr_F-t* t*
f*t-f u(x,t) ;
( \Yo jrr "[.ai
J:
u(xt)= + !** flr)[l- ,'"n't ,i(wx-wt') dw]dv
g-i ,-/f-r.b; JlFl r(,!-l \+j reie=cose+isine -[:l J-f t.,ll orLi:.,,| b
,-t-rr-l {++ .:-,1 )f W !
u(xt)- \ t f@tl ,'ctwltco,s((x-v)w)dwldv7r -@ o
,t'st 611 .c-;lrt, jtr.rLi:^:, b.r):L. J- q Jlir;;a ,.fti,J:rt, ,SJr.,
.g:l>l€.r\i ,-L'+ JJ u.J-s d:"- J- q dJ-!
-r5J-.rA)'Lry,JJ 1.lt t;&'-' .f Jt':'
t tt+t'Lt
c' L{uol
u(ut)dt= # L{u(x,t)}
\:-i .SJ.r>t) "t',.r+
! .J,t
Ilu=Crltn
I sint i o<tSinu( ",t) -f(t) - I
L " ;F-rdLnt?Iim u (x,t):" ; t2 o
r+€
u(x,"):" It,(x,")=" ;a)\-'
L{uul:.r'L {uI-su(x, ") -L{.ut(n ") } =@
Lluol= f *,-"
uodt= # f* ,-"
st7
fl*rlr V(x,s) = L{u(x,t)l Ft\S'V=c'Vstr -lvo- V=o
'/-f :lr 6bl:1 .L-,1 J-r*,Ft r,-, rlrL. 5y d
,)y Oli;*' ! OIrt r ..... . .... . ( t Yt )
V(x,s) - a(spk)tt + b1s1t(s)tc
C*t-..r F(s)=7{f(t)I ,Fj e sg,tetf b; q qi U .:.-,,1
V( .,s) =L{u(.,t)I =LVQI =F(s)
lim V(as)=lim I e'8 u(x,t)dt= I e4 lim u(x,t)dl=.f+O f+O c . tr.a€
ta(s) = o {5 J5€r./t.n1l Lf y Al s
V( .,s) =b(s) =F(s)
4+1:j J)
V(x,s):F(s) t(s)tc
u(ht) = f(t -it H( -:)
,9,$urtS-..Jr-+ V. t a = :gl;l q J ..r"*) .)L-11 JErt lJ) +i3 t .?i tt
c*t-r
b-J
trf,l ; L alaL +tn
,(r-, L5t^l+
lsku(x,t)-
t . ,
t-l.ll c,.-,,r qt\ tj d:..- .:l-r+- \
Ltu=uniu(xro) =sittx+sinYx, u,(x,o)= o,u(.,t) =u(n,t) = "
. f o in*\,fAr= v lirfro+rnyx)sinrudx=f {7t c trt L t ; n=\ !n=r
oJ.i ,F JiL-".\P.Y
o{lJtl
( l Yv ).... " " frr Jlai
u(x,t)= sitlx cost + sinYx cosYl
'J=t! lsuvv=o Jus=' c',jrL' ot('-r '/'*:l-ra-
Y
t+,i ;t Ju=xQ\) +tP(Y) .*!f lts= o )l
uw=xf" (l) *!"'(Y) ="
., lJtl
0, 0) =" J ,1,'(y) ="
4+t'i J)
tp(y) -cy+d t Q0) -aY+U
J{ltl+
u=x(ay *b) +cy +d - ary *bx +cy +d
.+t"t lrur+ut*x*y* \ = o rJrLr '/-P:l-f -f
:lf ,-;1- jU ip.l-lt d:L, d-t' :lf ltn*tt*=-(x+y+ \) g*;lrP,,4-di ,) r3=-\ ttr-l Py+p=.
,rOU-,-/ ttx=P ,f-f lr..lOt f lrun*ur=o rJ.:L'J -p
otV py+p:-(x*y*\) drL, e-te :lr 6.:P-lt(x)e-vii ;l5In ft=,€-t' :lr.sl-rl, rJrL, d,Jl q+ s> sP=-@+!);l .:,*^,1
p=h(x)et-(*+y)
g;lrP =u'= # ^s,i q +i! 'c-"1
# : h@)e-t'@+Y) r+i;r.r
u = e-r t h@)dx- i -r* tt,g)- Q @)e'v + !)(y) ! -,
-r-,tr f lf .J.ll r*ilFl *r(x,o) = o , Ltu = c'Ito.S.l5J- .J:Jl- ssb Cf d:"-- f
t
,jy Oti;*, ! olrLr ( I yA )
u(x, ")=
+ ;.s*=+
' (-t+7t ):Lst.Y'\o Y Y f
la'o ,+'.x<n
o^= + I. f(x)sin nx dxnftx
-suxrudx*\o=LIT
lvo
['n'' (+a !! )sinrudx+ I',tlr lc lo rnlt
Y ttn=-su?-&tnt f
u(x,t)=rt=, # sin+
F' vG'-=1--=pFG,Y
G,k-=-Gyr
,Xtt( \" "
)sinradvl
t+i;t.r
cos cnt sin ru
'J;5 ,F C..e ')u u.lru*=Yu, d:L'- 6
\sF'G -,yFG' '.*t_f
u(x,y)-F(x)G(y) ,-i-,rU.
F'T=k
p rll .ll t t-; UI " ' 'J
( \ Yt ).... lJr sl.ei
d/lJt+ \:-,1 G:f sF:aF ,-/-* Lthl5 dbl) .rt'i r., 6u(+y)=q/'I
.-r;5 F ,;.* Jstu.lsur*u, = YF + yyt rJ.rL,-2
C*t-f d.rL, .,,,,.li .r)-S d/Ji-t? s u(x,y) : F (x)G (y) r-i t \F'G+FG'-l(x+y)Fc
t-.r
F' G'1-Yx--+*ty-16FGf )L/y; rJ:L, J) t t+T;l-l
G' :Yr-k , F' =lx*kG'JJF
-11 .ul.i.;Lr .ii.rL, Cr-l cShl-r+
G:d''b , p-rur\ +k
d/-lJtr
u = ce'rT +YY + k@'Y)
.15 F a-r-,)s:u.l2uo*uw=. drLr-VFGU, =
" - K U_ t F'G *FG':. f=:b_./ eJrL, s> ,u(nil -F(x)G(y) .lrlr-;lj U
e\lrcgl:l: 6 G' * 1t'G= o J F - p'F=. f*U-F k = F''etl.G=c cos py*d sinpy J F-ocoshry+bsinh px
tu(x,y)-(acosh px+bsinhpx) (c cos 1ty+dsinpy)
g*trf k=-P' csl-rl q )jhb^ tu(x,y) - (acospx+hsinpx) (c roshpy +dsinhpy)
t+i;t JG'=o tP=o t---J r-11+#k=o dljl q
u(x,y) -(ax+b) (cy+d) 1G=cy*d , F-ax*b
uiS oli*' ! 9)rhr
f--llr.-tJJ- l, xury = luw * n, rJrLr z=ry tv:x .P .# t qi q-A
Ilx = Llv vx * ll, zx = uv + Yuz
uy = urvy * lltzy = xuz
t
Ury= @, *yu r) y= ury+ It r* yur=u*v r* urlz, * u r*y (u*vr* tt,Jr) =xuuz+ uz+ryu zz
t
u),y=xtlto
C*t-f +J:L, .r: .ib!" dr,,t .JrJsCr';db tr
xr tt - * xu r* x' yu o:x\ yu o * xu,
-h ,r-feJrL, d/-lttf ; &-,1 u-OQ)+ttt@ d-f :lf cEl.rl: tSlt,r=" t ;i jlr
..:-,1 u(x,y) -Q@)+1P(xy) d-f Vl-r+ dbl)
{>.-;)tr ol.l f f.1: Lr 6 r1ltt'.lr J*l* lr.;.+L ", J:b t'l.a '5''-.,1r Lr-{
?F ,\. l?galcm' Jl+ ,\cm' P P !'r'r ,-*jl 'I.' t o)J tit'a oitz
.ri;q P.-r^iU "|.6?callgm de 8,jseVJ)\l', #VJ'",rtu! c, .l ,r.ly.4sl otlf aat) 1 r:*.,1 ,-te;Le >t'l5
f(x)= x(\ "-x):.' f(x)-sin "l\,* :..eJl
f';lr
u(x,t) ='
o"eA'nt sin
\/.fN \"1?x"1"52
(\r. )
ctk=-
€
n:
+{
nn
-x\"
= t /vOf
, C'n'r'rn'=ffi="/\vfn'
''grlrin Yxaxan=
6\af@=sin'/ \x al;l q
a,= * ['' rh.l\r,,r*+xdx= { ;;;:l\o ' l' I l"I
cr-r.rrtr
u(,'t)=1s-'tvrtsinftx f@=x(\ .-x),Jrjr q
o"= * I.'i(\.-x) t^ +x dx= #(r+1-\)n*')
t*t jl
a(w) - Y I' fg)cosua f,v = J- 1 "f t
x costlu dx+ !' {r'tlcosuxdxJlJ.fr. I
=#ficosw-cosyw-\)t+i;t sb(w)=' t
u(4t)= * ,f '
* (tcosut-cosyw-l)coswx e"t dwlJ.
u(tc,t)= f +( t+ (-l)"*') r''ltwnrr sinffx,=l ltl
Jifa>..,) Gtf.rr.\rk l2c=t .Sljl q /t::ltr tl.'.5(. 2t '>2lr>4>.Jr- \'
?,. . (.r( t u r!t{
'rlr ql'tl
f(x)-- {T-.r,'\<r<Y ; f@)=f(x)
|. . ,' r)Y
u(x,t)= .[ *
@@)cost+u+b(w)sinwxle'n" dw
jy oli;.L ! glrter ( I ry )
.rb..rf 6 4taj -L:k 12a=b= \ ,r*t./ 6,Vb 4, r) u(+y,t) .ilFl-;l-r.,i..- '1 '1
t{ rJ! lll4sl .ilFl t -k l1t ,>o- tP.ri g;tod 6j,f&y)E 6.3 \ xy ( l-r') (l-y')
(t';lr
o6
u(ty,t)m=l n= I
sL,r. = n tlfrTl;/] di lr 6tt
onn=Y
(t'*rlr
!.'"(\-r) sinnmxfla - #
anm=\/ff #ta,i:lr
Otl,1q b^n= o tu(ny,t)= # i ; # cosrt,l{+ft sin ntry sin m n x
m-l n=l
y; rL..- J'- .rrlt .t fU;Us=Xi c<.f<\ ,l)o
u(t.)=Y,r; .SrS\
u"(.,t)=t; u(\,t)=tt ,t2o
gtc.J-u(\t)=v(x,t) +w(x,t)
eJd )jhu p-l LSVv(\,t)=o tvy(o,t)=o dpoL_ d c;t* l",w
(\rr).... ptJl..l
t+i -it s w (x,t) =a (t)x+b (t)
a(t)-1 t_ t=.+a(t) l_s uy(",t)= vr(",t)*wr(",t)
b(t1-;-, t,- tr=o +t+b(t) t-r u(\,t)=v(\,t)+w(\,t)
{+# J)
w (;.:t) =xt * t' -t= (x- l)t +t
tu(qt) = (x- \)t +tr +v (x,t)
u(+ ")=v(4 ")+w(x, ")
v(x, ")--1v t_.1 \x=v(x, ")
us=vn rUr=v**t g tls=Vs*wj=Vt*X-\+Yt
f-llr lt -ti dL-- {++t J)
vrvn= \-Yt, v(xr.)=lx, vt(",1)=o , v( \rt)=o
e:-b trF Cb v(x,t)-F(x)G(t) ,_e.f t tr.r..u.. .<sit ,fs: r, rL.-, al p cSla.
l5ir- y; d:..-;r r5p1L-r.
F - kF:o, F'("):", F (\,;="
Jkn = fin + 't) +.li;r lf,Fr,,(x) = coskr;- .--,lle,Sl.,,lr d:.-, o,,l
Jr-Sdili-r+ q JJ,SI g. r /<=- ktn=-(ln+ \)' +
6
v(x,t) -
C*t-.f vt - vs= \-Yt eJ:Lr;r
jh JV-'^ ! o)rtr,.
v(4t)=
(lrf)
Gn+ktn Gn= r .[ t(t
-lt)cos knx dx =
GnT=Ane'k'n,. #- r* *€
tl= a
*,o"sinkn-tlAnifr=t
(An e'k',{ . fr. ,* * )cosk,p, i kn= (vn+ \)
C*t-.f v(x, ") =
t+,T;l.r
cr-t rt+tt
Y
Yx q .?i tl
|.rn"+*)coskur=rx
I
I lx cos knx dv =
€C o..f ortrtr
U-1
An= *n ft((-\fk"-t)-c,l
u(\t) =(x-\)r+rv .:., * ft k \fkn- \)-c,le'k' nt Iraa3.2: .p
dh +*I cosk,,x
JJ Vl-f,&-, J- c'"t yjlt .\f J[lruoluO=x+Yy ; o(.l1tr ; c!1z
u(xr.)=y , u(tcrr)=l tcrlstl'u(.ry)=y , u(rrfl=,coV ,o3!3tt
r gjlry eu(xry)=V (rril+w(x,y) stf I +ll*.-, cSlt Alf 6,t4;..'+ q .Jl.>
Vj,lr,& al clt.V(",y)=",V(",1)-. .rf g:1\el ct*ltw
( \ro ).... " lrr Jl'ti
w(x,y) --a(y)x+b(y)
b(y) --y , l= " +b(Y)
a(y)= L lcosy-y),cosy
fi
t- r u( .,!) =V( ",1) *w( ",1)
= arr+y b- : u(n,y)=V(r,l)+w(n,Y)
u(xil=
w(x,y)=
v(t")= (\-
V(tn)=* (t +rr )-n+r
uw=vw-
Vo+Vr=x+YY+ l rotYfi
V(",y)=,. , V(r,y)="
u(xil=F(x)G(y) €.f \
4?::, J)
t
t-t-,/ 4+rii u(t.) = v(\") * w(+') )l
x=V(x,4+ Ltl
t*
\s u(tn)=v(tn)+w(an)
ttn=Vn
f-rlr V CVls ti d:'-.. r'?di J)
; v(x,.)=(t- I )x , v(nn) = J (l+*)x -a*Y*tJ
d.l4 Jlt'{ iU i* il.l{ d:"- J- q r':'*;'i
€t,lF cSl,;, l; l:-'- ar r g,jlr i€
x,t
x,t
(cosy-y) +y+V(+Y)
(cosy-y) +y
\ )r b-e,t
x- cosy,l
Fnfr;)=sinru 9 k=-ni t-*,1qr$ .li;l rf
F - l<f=. ; F(")=o, F(n)="
Olt' t C)f 6,J4;--+.{ Jb
L;h&n'.'-!&)rtl
V(1:t)= ! G,,(t) sin run=l
C*t-.,r Vo*Vr=x*V+ * cosy )t.li gtr tr .g;lry Lr.V + )ry.l l:-.- 6lr,
o
nrl tt
d eot-.r cst* l, G t'5,i -,11 3
Gn-n'Gn= tL !
-&*yy+ JL cosy)sin ru dx
tt.r3
{.qr.:i JJ
G"(t)=an coshy ny + bn sinh ny + Gn'\)
.l (+t* lt Gr'(y) ) c--l iy ..JrL, opr.ar:l-l+ &-Gn'y).li;r 6
v(x,y),l=l
€ 1.*t-.r 6blf b-lt:1 {+i t{
V(xr.)n=r ,J
V(+n)= i @ncoshrut +b*sinh nts* Gn'b)) sin ru= t ( \*n)x-n*ln=l 1l
t+i:l-r
an+Gn'(.)= * (\-+ ) ! **
sin nx la - J- 1\ - \ )(-\)n+,,t ,t fU, ,t
( \rv ).... ..lJt.1|...i"
ancosh nn + b,rsinh nn+ Gn'fu) = -l- (\+n)(-\)"*'+11-n) I U-\)"*'+ \)nfi ,c
.t;*rl r-rl' bn tan t+T J- t{.t
yj +E*" J- *l -r.;JJ.,.\f J[lrua=uo*uO*x+y+t i osxsn oSlS,ts t2.
u()6y, ")=x*l!, ul)c,l,c)=lx-l
u(",y,t)=y*t, tt(ql,t)=y-t
u(xro ,t)=Yx-t , u(x,.,t)='-11
dfot-"f ,St* \V tw=ax*b s u(x,y,t) =V(ty,t) +w(\y,t),r.-fq .;l-t+T ;l 5V( ",y,t) =V(tqy,t) =.
w-- L x+y+tJI
;l c,.".1 crt!" V ,Sll4l.?"-, 9
Vo=Vo+Vo*y*x*t
V(Xy,.):y*x ; Vt(qy,.)=lx-!* Yt - \fi
V( ",y,t)= o,' V(n,y,t)= o
V(x, .,t) =x-Yt * ll t ; V(x,*,t)= t5- tl * l! ,'ott rlrb.rlj \,"€nr.rti:.,1 tut,&+;l .1;51 ,'r
H:F,(v) = + ! " r&y,t)sin ru dx
p*t-.f V u. by.t drL, tl ,yt.S uty .!-r+r
Ha=-n'H+Hw+ I t o
6*y*t)sinnxdxtt.
it\e=LI" xsinrudx= I (-\)n*'
Jt- n
jy or;;* ! &).rh,
n= ! I " sin nx dx | (r +1- \)r*,),t'. nr I
('*t-..rHo * ntH-Hr=n+B(y+t)
H(ryy,o) = By+A , H,(n,!,.)=(l+ -]L) A-(y+\)B,3
H(n,o,t)=(\+ !) A-VIB
H(4n,t)=(y* !) n-(rt+n)B
f=or.,r rlj dF? JbH (n y,t) =Q@,y,t) * R(n,y,t)
g,,-l crr-2 :l.tl 6,lrr. Q@",t)=Q(n,z,t)=o rSC*t-.f S:* lrR(n,y,t)=ay*b5
r5.:,-rl p) .L-lfR(n y,t)- I- (A+Bt-Bn) +A(t+ L)-rat
't .ot_1'l i Jj O_2;e u.Q Slt d:.- .11 q {+j t{
QrQyr+niQ-,4+(y+t)B-n'[(A+Br-Bz).I- +.e1r+ 11 ) - raq
Q|ty,.)=-A L + lBy
Q,Ot y,.1=1A-By+ yB- ' ,Q/t, .,t) =Q|r,n,t) = o
n
:lft.l r:tl:J, dL, J- q ,:li',dl-, Orl.rr Q= i G^,;in my t;>ls.llj trm=l
Qit"y,t)=_:, {E*rsittt,*,t+Fmt cosr,rnt* h I B*-n' (A +B:-BtrlJ +
.r:-.1
h [An + Bdt-nt (Ar + lAt-Y Brt) ] ] sinmy
,= j!- (\)^*, ,D= L lt-\)-*'+t/m fnr
E,*,- * Seoprt'*/z')+BDr (lm'-n'-\) qBctt *n-Bcm'\
F^n-- # VDr(n' - l)+ (m'A'Bt *nt Br - | Brmt )\
,.li.2r d
.;,tlJl'f
6€
+>n=l m--\
u(v,y,t)=y*t-y +r* 3 t! U+Bt-Btc) + A(t * # )-vn* sh tutt r-l tt
{E,^Sittl*rt * Fr-cost*rt. # [ Br-nt (A +Bt'Br)J
. # [Att+B*t-nt (Ar+lAt-YBtt)]] sin my sin ta
r;fJ- b ti Jit -, jl 4, r'r- \
a. lt6-crlto=o ; u(x,o)=n rut$r.)=Auntr, u(.rt)=u(r,1)='
13Jltrt gtl'l-J*t.\V.Y
'r-/lJ?O
u(x,t)n=ln*t
b. uo * aq * bu = clt'ts
u(x,.) = f(x) , uft o) = o
u(",t)-u(\,t)="
lY/G t)n- t I sh nct sin rarcnt (nt-l)
,.(.tr(\ , t).
u(x,t) = an GnQsh nax vlr
I [." flgsin rua dx ;)1;r rf
o
ttr I
JOn=
[ ,'" (coslta+ # t* at) ; e) o
Gn@= J
,'ath(t+ 41 ) 1a=.
I e'atn @osBt+ # tn f, i a1c
p= + ftQ+ntev'c'1-a'1'tt , e= { fotr (b+ntntct)Jttl
c. tto=gruo*sit*a ; u($o)=. , uttn,o)=. , u(.rt)=u(\,t)= \
u(\t)=V(+t)+w(tc,t)
olV(x,t)= ! t-f I wQqv)sin ruvdvl cos nct sin nax
n-l t
w(\t)=-c' sitrh x*(c1 sin h \)x+ \
d ur=luo ; u(x,o)=x' (l-x1 , u(.,t)=u(\,t)=o
'YlJ?
.lT;l d
.vlr
u(x,t)= # tr (\)'* t-11g''.tr" ,h rr* .:lr
,t; l:..- J- C-,rl .-r. ,J,Lr- V
u6-u,o=ya1 ; u(\.)=Y , ulrro)=x , u(.rt)=sint, u(*,t)=lt
u(rc,t) =v(d:t) +w(tc,t) ; w(x,t)- lt-sint x+sht
v(\t)=a$ost+b,sitx+ft + v ! t *n
o
n-l
( lrt
tlsltlo= fl' r' . <t< \ , t).
u(x, ")=sitttrx ,' o s.f 5 \
ur( ",t) --t, ur(\,t) =lr i t2 "
usk lts= xlost r' o (.tr(r , t7 "
us(v,r")=sita i osxstt
u(o,t)=11 , u(4t)=tl ,'t2c
ll; lts=t r' o (J( \ r t).
u(r'r.)='Y ,' o sr= \
ur( ",t)=\, u(\,t)=tt ; t>o
; c1X1\ , t)o
Uuer.)= o ,' c 3trS \
, uo(\,t)=cost i tEo
lll!.ltls=Jd,' c (.f( \, t) .
U(x, ")=1 , u{t ")= o r' " StrSl
us(",t)=" , u*(\,t)= \*t ,'tl.
b,= Y- *,on=+f ' ar=' irTrr rf
.7;,,Ft-, jl &- "f ,y s.rl .-,Jle.-f
b.
c{
I uo ,r,= o
d- I u,Qc,o)--x,
l r,'l',t1=s
e{
I{It6-\Ilo= "
u(x, " ) =sin
u(.rt)= +
,
tttY
,
c(f(\,1)o
, ul\.)=llx;.srs\ux(\,t)=lt i t2o
drL, .9q - .',yrJrL, f u=v(x,y,z)e-iu ; i= GT jtJt+ ! .S-r+t ,l[.:i- Y
lle.trvrv*krv=. i k
wc
d.rL^, ,S4lf @.1r drL, CV.lf jl oLi:.,| tr r 0i q lo.i1u. s t-rrl JeV
.,/jtl'"""r,':t:fiuo--c'(ur* + "r* + uu) ; .<rcR, o101ln, t) o
u(rrL")=r cos| ,l\(lr0rn)=. ,' osl'SR , .s03l:r'
u(&ert)=., u(oril,t)=. ; os9sltt, t>-.
$ ryt-il &- l rY '.:'-"1 ,:)L'- 2
I uo=t'@o+W) +ry sint i o <x<rt , o !1z t) o
I
l"W,o)=o ,ufty,.);.strsd , oslsn
I u"(.,y,t)=. uy(n,yrt)=. i c;;lStt , tz-o
lryr+o,t)=.; )(x,n,t)=. i c3x:=tt it2.
I ur-t(uo+W)--x+y+t i o<x<n, c1!(tt t t).
I u6y, .)=. i c 3x3tt t 'slsts{| "&ort)=o
, tt(xrr.rt)=. i c3x3r- , tz-o
I
l.u(.ryrt)=o, u(4y,t)=. i osxsn i t2.
b.
(ltr).... ...ir!Jl'.i
luo=uo + \ ur*r*t i c<r<\,1).
c. I ut,o)=r,'ur(r,.)=tr*\ i osrs\I
I n(\,t;=11 i tz"
I uo=u* i c 101tt 2 t) o
d- I u(o,.)=g,ttlo,.)=sinoiosostt
1,1',t1=1 , u(r,t)=rti l='
d.r,11) =ytf(.,-c\rf(t)r-(+)F,{ft d,-r0L5-v
b. F,{fI = + [(\)" f(t)-f(")1- ( ff r F,vI
l=* .ybl+t
F"lfI - Y [(\)" f (")-f (")]- n' F,Vlti .1; .1.- t-r/ .,l,--^+ ,JJr, ls ti,Jtt--, il 4,1 - A
A. tt1=lts*X*t ; c 1X1, 2 t) o
u(xr")=yt , c1x1n
u( "rt) = o , u(a,t) =t i t) o
b. u1--tts*x*sint ; o 1x1ts e t) "
u(t.)3c ,' c3xStt
u( ",t)= " ; uxfu,t) *lu(*,t)= "
c. ttolcr"go "
o <x<rs 2 t) o
,)Y otz* ! gIttrr ............... o...................... ....... ( tft )
u(xr.)=. , uft.)=o i 'SXSp
u(.rt)=U(rrt)=o, tts("rt)=o, us(nrt) = sint i tzo
.rJJ- u_t cr$_+ &J+, bYd eYa '>l;'; e
+k u-D,-l',--^t, ,SJr, ls tj rl(r;-.r rJl, , yb? - q
llslt4r=. r'-o(.tr(o r l)"u(tg.)=f(x) ; -a(.r(o
.J;Ss, ,b t* c.,..-, q lx I ... 6 ;p1ux Su tg-rl ,|xlfl-- S sp:u
uw)=* I_1ffi0, .vlr
+t t b tj,yj.d',.,- Vl3-.- \ c
Us*llrr=c I -o(.tr(o t l)o
ryQ6")=g(x),' -o(r(o
lul.- 2t\ua+o
y+a lr | ..o
'r./lJ?
u(x,v)= + [o', {-ffi tv ,.c= + I--llour W ] dt+c
! rlstdl-l q J,tL- g1l t-T .J5.,| trr.,c,. .Js,it ,ist llt tjJt -r;l dre - \ Il.s*g.*;.tS-Vtr-
4. llg-lls=X*t i o <X<tt t t) .
u(x, ") = Yr- I ,. c *3n
(tfO).... frr|F.r
uaQr.) =r+ \
u( o,t)=1t
u(4t)=1
b. uyYuo=x-t ;
u(4.)=cos ;
ur(.rt)=t ;
uoQqt) =t- \ ;
i cSXS,rI
t2"
t2"
o (.tr(rr
o 3X3tt
tlo
t2o
,
,
, l)o
.J+tJ- ls J-s JL-r;l .i; - \ Y
lloa. lls=llnlur*lt i o3[Stt , csl5,lt
u(Jl,yr.)=y, ulql,o) = o
u(",y,t)= t * y, u(o,y,t)= !u(xr "rt)-., u(xrart)= x - t
b. u,r=yo*uo-x i o{sn
ultt l, ")=.U(",1,t)=. , u(41,t)=Y
u(x, "rt)=11 , u(xrr,rt)= "
r .Slsn , t?o
c. ug+W*tto=3-7 | oSXStt , cSlSr t cggsn
u( c,yrz) =2, U(nrl,Z)= o
U(X, "rZ) = ", u(Xr*rz) =x
u(\y, ")=y , u(x,y,d)= o
,)1r.oti;*'!&)rbr . (tft)
d uo - @o+u1y) = cog cosy cost I osxslt t o3lsits , tzo
u(xryr") = cog cosy , ulr,l,o)=o
u(";,t)= cosy cost , u(a,y,t)= -cosy cost
uQq",t)= cos, cost , u(x,4t)= -cos cost
rJJ- ./)tJ .J,-r+ ,.(.5r, l"t tj Jt .- jl 4, J^ - \fA. llo=g1 Uo+fg io(r(o yt).
u(xr.)=" r'o5.r(o
ulx, o)= o ,' o S.r( o
U("rt)=. i tz"
ur(x,t) -" o$liTr*- .6y'
b. ttr= urlu ,' o (.f (o , t) o
u(x, o)= " ,' o Sl( o
u(",t)-" i tz"
u(x,t) -"
C.ar tlo=Y, ,' c (r( o i t) o
u(xr")=" i o5,r(o
uy(.,t)=-1 i t2"
u(x,t) -"
a. un+u,--I-l ; .41z- i "91n
.61^i1.r*. .6.r^
.6liix*- .6tr'SJ-Ut;'Jtt-.- \f
lJf JJ
u(x, ")=. , u(xrn)=Yl
u(",y)=lf , u(n,y)=o
b.tlo*lt'r=" ; "4x1n
uy(\.)=Yx-\, uy(Lr)="
ur( ",!) = \ */ , ut(r,y)=l ,' . sy*j
C. U*+UW=\ , c1l1n ; o!(tr
uy(\o)=o , u(x,n)=sinx ; "1x1n
u(",fl=sin lf , uy(n,y)--o i "91n
dV'It=o ,. \(f<1 , o101tr
u(\,e )=sine , u(l,o)=" ; os,osn
u(rro)=" ,u(rr*):" r' \srsY
e. Yrtl- c ,' o (r( \ r'
u(";,2)=sin8 y sirtttz ,
u(x" "rz)=u(\\,2)= "
u(x,y, ")=u(\y, \) = o
f. vtu=o; r<\
u(\, 0,g)= cost O
c SXSI
. sysrs
i ,91a
i c S.XStr
c<y<\ ,.<Z<\
u(\,y,2)--.
;o!11 ,o(0(Ya .19(,r
Pe19@
-hh*, Cf-p
.bLi, sl.tcl .\.fb-txt =-f f*U- C € ,J,.rtt-f =o d:t-, J- -L" ^+t axt+f =. drL, J-.rt
,rr+-)& 4,.1T;ICT ,r,+ * bb l:+s.:Je e, V6 l,:'+-St=+y/{
,-iL-: '>lsj+3,,;l irT i.s;L ,ltf fb ty; .:*,1 ,?V t:+y &- rf J.l .._,9
)& 4, blf q ri b U Olttnj r:-,,lxt+f=o rJ.rL,e .1l-r+ A Joi)trJ.Ji=d.r
d-i, ttf aA^. ut'f-f )Jt 6- Jl-f c.'.t ,-eL; t-r+-f 4,.llf q {<! &lS x ,*.t- rJr 1J J^ 15ljl r., ; J-.r.r,rr ,f-L; i r.,')-p^, b 6 .rb .;lj ef U t!f:i'- rl-tpl cSU ,*-f ffi 6 -L-t,-/ )'t u.q;.bb.r, -tL;r:-tr ...i- lrx*iy
JJr 1) e )J.p y' lJ, L!,J +, f"-lf fr" .t"tta J.l .r-"tr ,-,uLt grjorL )l '*+f ,"L,r c{-? O** .t.S-ul.r'- Jt,fut rJy o.rrj 1gt4u., rt F6.l) .Xlr c*.,1 tL;.,
,-i'gs (xry) jt gDol:e r, ! JaLu'., .aie 2tz LL:>' rJe 4, gfj-r.l;l
5Rez--x L,-yt4 I .Ju,b,-/z ,/g-f {,".-.i ls! S ,rr{.je-.}.'-.i ls x *t:54
ll:i, rl-t-cl lc-2.eu t rf .L*'1Ut Jl J.lte +t L;,laet -,t \ ,rJ 4S Inu=y
.tl,-,| lyyG={6,fl I x,yeR}
u.(. ;12,,=(x, ,lr) tz,=(x, ,l) P s> -p ,Sltlt ?-reJ C:? j*e 5t ^?.rt+
.fnll5*,*r- Y-i cgLrc/ tfz r*zr= (x,,,1 J * 6r,1,) = (x r*xr,! rfl ,)
tz&t:@, ,yJ&t ,l)=&{r!Jr ,trt)/r *xtyr)
Al P.srr/ JU=. .5,- J$:i (C , + , x),?t' J C- .f.r er q C .Kti
giC dri? g.+ ,-ir..; ! +i b .,:.-,1 (l .l . ) iri ;t<,- PJ (oJo ) oK:."r
(x,y) = (x, " ) + (.,y) = (x, . ) * ( ", \ ) (y, . )
rl-trl ;l 6|ro3-a.J) t & rl.tel re2.qv ga 6_ q ,5- ,S-bA r*T c^Lr +,
! qi tf .l r.ltt5 u.ltx :b Fs. ) cStaltli C.:-,,1 j-pl6 . ) 6,1-;el t tLi,-l+ij Cr-l r+ CVs* d;.3 tr ri=(.,\) g--j _/ (",\)t =(-\,.)=-t +3-l
rc-.-,| ,alr1x+iy\(x,y) tLi'., tJo 6 gtct'(+y) =x*il ; i=( ", \)
r gStt+;l dti t ,-;u,l tSVta r, -rrtc, 4|u2ja, o6*-.,r .4 JrL.! 6llo.)-a
l.;y .t-19 5',5" x sF t bl'...'=rb.f dJr-f t & 6,1^U^. qj t b t+i
JJr y' .p=lb.f l"l;i, .^;4 4 b c,,t^a:;., 6,1-,.ya.dr:.? t 6lo*ir .flC'i.c,..'ly Lr.-f $ J* r, 6l&.8 z=x*iy lLi'.,
rl-tcl.lr ga-i .J.rr.rr "f-t-7
u. s oJiab z--x*iy ettl tLi., Irx-iy tLi. r-te
gr;rlrzr 521 Wu rJr 1) gl;l q ilr.r, ..-:*,1 ,i'rtt; JaL-,
z, _ x,tiy, _ (x,+iy,)(xriyr) =
ry'r*)il, +; rd/r-rr/rzt xr*iy, rtr*},t, rtr*J/t, '' rrr*yt,
..:*.,1 JaL.i,., 6.:-r.r rf
C+) rly '6r'y=r sin| t X=r Cos|
(lo\ ).... lrp
f-rl: '63iz =rcose * ir sin| = r (cos| +bin|)
6 0 tsl;rlzl=,/frT,J:.r.:,rr.r--L; lzl t.l oJ*b z L)s'.,r.rr.lrJo, stilrr
fllr 1 r:-,,1 lyyz r-,,L-i t- it J.li o .:-,'l oz 5ox ,n,L:li
Q=a77=arcfunLx
t?t= , lz&,1 = lz, I lt, I
* =cosl+isinl J:*'j6. -[:l :l '*-* *te J-rbr.. Jt^l J|tti 1.,-n<0=* 5
p*t-.f J-rt cr-l q qi L .c-,1 lry E .[:l J-rt q ,S.rL; crrl
t=rdo
.r-5.-rB ;lli,-l JtUr',..:-,,1 lt(rti i.lrr ! ?sVde
z, = lt /@r'ei , z&r=rreio, rreio, = ,rrrdQttor)zr r1
ltrr,l >llz, l-lz, ll ,lzrz,l slz,l+lz, l
( cos 0 + is in9 )n =' ss t r0 + is inn0 bsi-r Jyt)t-,\^'sittx 5cosx;l jl#l1.,
-J;tli C l2sin ta 5costtx ,r-l -r' J:'t cgU il
olc"t.SJq, rt+'f otre pU u. )4 J.fi'., rl-tel tl sj*-:.)J l
riL oet* 6t-vz:rd0 +tS,ft ,Jb, Al Slt dil).;/ crirL, J- q ttlir.l
)ri w=z; or3-. q b !i 15 v{=z dCoto gt* €lr,S lrw=pdQ.o6i;i
,*t_f v{=z ;l V,.t,-f
lrEi, Clt. ... ( rOY )
pn{4=d
, o+tkt Pn=rtaoi;lj
e-- r'" , nf=O+lktt,ntLr'- ;Jl^elz;ln r-1, d/_lJt{
..0+*tt.l-t
Wk=rlf e n t k=o, t\r...
tj Jt* q *+Jn sgl.zb w:z'k d/lJt{ ..:*,1 Sj.&*.li "*rrn !:, rf .:.-,1
n- i10+*tt ) *l
wk={r e n ; k=o, t\,...,n-\
.+k l.lxt+\i=o d:L, gklrr leLi, rl.r.el ,,:b-.":r;;l olti:.,| lrr JUrlllf q
(,-rlr
rt=- \ p= \ ?d" = 1 1ti(tt+*a)
t+T;lr
.. ,J *ttl(-+Xk=le t Y ; k="r\,Yrt'
i(L+n ) irn in 4+,sJ)
xr=le r r' = le, =fiF\+i)rx.=le t =y'111+i)
: Ylt orEa-
xr=le ' =y'i(\-t) txr=l€ v -/Tf-f-rtgb-J-t.Y.f
a) Ilz, I-lz,llslz,-z,l <lr, I+lr, l
u:W<ptstxl+lyl,
.f .t+r dLr.\
nl I zr | <. lrtlvl I zr+2, lltrl - ltrl I
d) lz,+2,,1 '+ I zrZrl ' = l(lz,l '+ I z,,l')
.JJu,,t-, b tj drbtf ;l t5,'r Jlri. .Y
a1 {-i u1 Gt/-v ,)rA d)fu.J.t+r JL.:j .f
1+z+z'*....*{= \-{+'\-z
.t-r5.-rti ..li r, ,+jL. s
\ *cosg* cosr0*...*cos n0= I a sin[(n+ \ll)01 ] o <o<lnY $in( 0lr )
oKsiz*\ e o.:y r-lr;ln 6br-,*.1 il ,-#z Jt .Sr+l.lLs.Y
\ +z+z' +... **-t = o
.LLi, o.i-ra f 4>l.9j.f .flt-r.l <" ,S.llr.-rb;r lfrl,,l U:j'- b;e jl ,,pe o2qttz..Li),r('r-t...oc
.S,jl-"
Ndz)={zlzet ; lz-z.l <"}
. D,rl;lr &,it.4 l;z.cD r.t.E.-uq Jefs., .^ie;l 6,1+ej.at.tt;D J;S,ftls z. tLil Nr(2.)CD S cst* q -t.iq sFy tNs(Z.) sz. jl a;t,rdt'.--r o6-P -r:,.ti
.tJ-iq t7y ,SNr(2.) 06y' r:".b D u+tV &4.5-
Nr(2.)nD=Q
N"(2.),lr-t-., -f gljl q o6y' -r:".bD 6le,lf $it ,S' lrz. 4;6
N,(z)nG.-D)*0 t N,(2.)nD*0
!!i,cri. ... (tor)
4 .-tJt-t .rblr I'Li, Ji lrLA ftr o6y' -u.b ;tr tesre.v 4 tJ D +e-2.a.
o6y' -u..b orL l1D tu 5-a..ri,b >jz G)l }tn JrLj 06y' -r:,.b f,*t b &j1o1.2 .li;l rtl; 1rr"o .ll; .6re r;,.b *io l2D e2-au.J.<J C- b r3- rlT iy P
.J".rb D i ilr €*(.i b Ul Jr!'ii qSdt-f P: g. cfu* €-<i Jar &- Ll
ILi, Cly.f .fw:It+iv lals,., >t-e z€D -.r,Sl)h.t., r.iLt orL Jf f^ 5;\t4-a'..dDl5,J.t.J-.f b {:il$ ):+s.ib'., tu 1)z -f 6\l q o6y' .urbz ;l .*b t, w .6lii ,tJ.***.
r5:.,11r z tw;a dl,!rr*-, 1l r:.,,r .li q ,yal-a:;l l"b, Cl-r lr L o-ur-i C..-,
J Clt lrw=f(z).J".:ti )J+ d-^, t: L .)JF.i dtd w=f(z).lqt.; 4 ! b t+i ,.llt
d--*,.lr.f=rb./ f ,if jo.:L .r* qUJ ,-;g oi7 l.;D orL rr---r 5 )l +e7-at
t- r-^r, &-ot- +JliCl, Cb -p ju-f 6g1rr5or.oL:, ,* Cli qJo*,'
2tw=f(z) Cb.c-i 29Jb 6.21:;-, ",qL;
;r:i" LLi'- Cli;) .)1.: "f-L; ih-, .5-
u t->i..o jlD' r=-b t) Cl-l w tLit 6_ q lt! sx 6;4;1 6illD q-b ;lz &l;r" il,
&tu 4,.rD.=-b ;lz=x*iy ^rA f .KiT w--z'.6y';{, S.f .I,-r* v.l
rIeA Cb d/-l 1 qt tt ){l. .t-r-,rlrl,,-r* w tai.,cJ)v= 14 su-x'-y' Lw--u*iv
w=f(z) Cb;t S ,;:c{+.rr*.:-r-, d ,}-;;w ., ;'--rr Yi ,,Li; +'z b;4;l t +i
Oy,e.li.1, t--.J ,1 \ww .EJr'1) 4ilu>b .}i lr u.z k^;e;l 6lr.2b .J'f-fl ,,
J,-ry l.z. tLit ;a ttw=f(2.) vle.l,-r+ d,- t- .:;t* 'Qw--f(z) Cb q
,y.*, -#.1.1* c{." P :tr71 ojl$l .h .tlq b +'-5],;8 ;*'61; ..r;".br, w=f(z)
ir*, C-r.-, j r.2l.r tSr;f.lK.-lit..r 11 6eV gtt.l .:*,1 lf ys,-r-J--r c-it* e, .rj
;U, 4;-11i:b- J..A o-rrj a;ta:^,.l.: .J.r:rr ,r.a[a:;l rl- q 12 e3 ar,ialjl Jtt
( roo )......... ...... lr F
*i &^, J.Li- Cli;l .r-.,rU, q l.l ,f +i .* sb'r^i '3bl iii- Cl.;; ;r 6
q 6c-,t4H t".r-l f oy.e f i.c.-.al s7f "*e- Cli.l, tei *t 'S f-llrrt
I )t*t13;t-tii -t ,P -,irr* 1r.,-,+J lJ:i,- Cli ,FX 5 'b' \f1) d'':'r
.g;lr; t+T
gJrtirrf\. ,-p-.1-,,1 w. J> 6,l21u,z. r,lai.lr w=f(z) gtls
Ve>., !ld)., lz - z.l<6 * lfQ)'w.l1e
.1, Cb JUi. 1ly. d.Je 6,l2ltz..lt Gb gr-l .61 13b r:";' z. thii 9l2fQ) gtls
u:^r- tit de.A drrl
lim f(z)=f(2.)z+2.
,.5c+tt bli C rfiU . .63i -u-!,q .r- L5blr z. .t>7Q) :f(z) .6y'
lim (f(z)+g(z))= lim f(z)+lim g(z)z+2. z+2. z+2.
lim (f(z)s@)= tim f(z)lim g(z)z+2. z+2. z+2.
tin I9 = gmf(z)y(rin s@)z-2.8V) z+2. z+2.
lim g(z)*.z+2.
r(;,-l n Jc5,-2t
!l:i.cr-i. .. (ror)
[7+. 6 isrycr.rt r o6y'.:,.-l .i*, Lgl2bz,&iirr w=f(z) g,V
.J..rr.rr ,-r--Wf (z) t.l o4tb f(r) eU .;: ., l.l.lle ,jal l- .J-!b -r- c5blr
f (z) = rim fQ+*)'f(z)" ' Lz+o Lz
csblr z tLi; lf?)=z' cb.5r.5c4tt.rlj,_l Jru o e,,br,,u d/_l t +i t, yr,r-tc; --.o ,sW=7 e3t CttJ,Ja+-r1r:-,,1 w'=n{-' J.lJw=* or.I glZ j:}t(:-- J Jn a- 't--;-
c-., j:.:., csbrr
e{l d-r-i! rr+ yz.;l .r(rL..".rr 4, .61 r:".b spsz.,rlii 1r lsw=f(z) gli.J-ib ,il.' gl.llr
"dt-*^ Orl J"LA gb.1r gb
Clt'yr.J-lb )J;z .,aie lrUi; q$.rr o6re -r:,.ti;b Cfr.4 lrw=f(z) gl;
.c-,,1 lb "*b .glrfq J:" d.,_ Otf q Cb J^ t- 7tu=zl
: g,*rlr ls tj:.lt tt..r # 9r j:-:r U JoL;.21.1r
f(z):#*,#=|r #*, fit
z=x*iy !;i.i :; f(z)=u*iv gb .6.rr .(dr--.1 ,r:S Jf l d:,At) \ 4;t.t<jT .r.i,b ;* gl-rb
gn-+rr
.by-f F*t.rScrYrL, b- \lrJ q.r)rL, gr_l
-r rll .gUll- .
0u 0v du 0v
-=-_ _=_
0y a)c' AJc Ay
(rov).... lrp
f (z)=tim f?+nft) =fim @+Nt) +i(vtlv)-(u+iv)fu+ o Lz A"*o Ar+iAY
AY+"
=limAtr* o
Ay+.
Nt+iLvAr+iAy
'.>-h5;;lli,;.iy -t- t7s cJt ^, J.l
Nt+iLv ,. Nt+i\v 0u . . 0v=.Lrj. =
-
as-Ax+iAy Ar-o Ar U 0xf (z)=lim lim
Ar+ o Ly+ "
f (z)=lim limAy*o Atr+o
\ 0u ..0v+( ":* +i ?):tdydy
dv 0u
-:-- ,0x|ya
Nt+iLv \ ,.= 1- .-r.&+iay t At*.
0v.0u--t-oy oy
du dv
-=-Arc Ay
d/-lJt+
.J-,-). a .lU-U- q qrai .rt."il.lrl) 't
Nt+iLv
-=
Ay
+Lii rr f(z)=u+iv gb iv tu olig.(df--J ,r:S l.r, ?et) Y q.ii
! r &-r* 6,il ;l .r(rL"*-r, 4, tr t $f o-t." ,?-s ,F c',irL. ,> z=x*iy
!.:-'.1 lltl )r:tf (l ,19-i -t^lL c.,,-l-, jf c.,l,.i::,
f (z)=#*tX-rt-rf j;.:,glt, Js.:' ! ..)[.il
f(z+Lz)-f(z) _ Nt+iLvLz Ax+iAy
Nt=uxu +urLy t e,,l(LiTTtEyiY
S €rt G.p.lt Clj + b*t d.r,1r
frLi, Cli.
Ay=yrAx *vrAy*
( \0A )
e,,l-lffiidlyA6AY* ' S ,-P:cr, 8, +' .litr d
flC q:$ JL--r,Sc,)rL'gtg-lL;il oti:"'l U
Nt = urLx-v r\y + e,,/(ffi'1 Ly-ir
-\,,rur= ! v,r
f (z)=
,,*t-.f i:.I cr,l!" .1r crl.l!" d,r-t .rrJ.(-b t
u c.-,,1 J.lJ s-r1 o22,e t, cb iL, t
Lv =vrLx *urLy * e r'/(Lffiit
f(z + Lz)-f(z) =ux*ivx*(e, +ie,)'@ .(Lv
LZ -wx t "x t let I wrt &+iAy
t Vr=
.r.rL" t*;i;t sLy*o ;Atr+. o63i fu+" .+tl7 Jb
l(e,+ie,) Wt=fiTTil
g*t-"f .irlJl+.rr5rrl;* ,b P r.b a.,
Iim fQ+ 4z)-f(z) = f e)=rx*ivxfu+o Lz
.bsu: At-t, q crlilr
.tL.a:;., tr dr-,*r ,* cr):Lr.t<ii f+)y=ffin| 5x=rcos| P -# o6y'
(ur+ivr)e'io
-(-t5-u)
.+it* )-t+J Jtf .trbf/)=zt Cb ,,i:'i'' .\ Jtl,
f;lrf(z) = u * iv = z' = (x+iy) = r' - y' + ilry
Iy=-Vr=-l! , Ur=Vr=1X t V=lXl ,Ir=X\'Y\
u,r,.Su.i.t." Fs.r5cr).rLr;.: to)U.u5a;tT if ir-rttfa tv lu,l-l1"\
f(z) =z'
{-rdi ;)
f;lr J o)t i* 6blr
f (z) -- = Yx*iYy= Yz
.r+5 c.^2f(z)=7 Cb et#: &-nr, .Y JU'
,v=-! {<!_l !+i! J-l r:..,,1 u2.t 
.+l* rfr-t o2ye 2tl.2fQ)=(Rez)t +i(Ina)t=r'*t/' 6b ,a-:, .f JU,
f-rbItx=lx rvy=ll tv=f' ,lt=x'&, , Cb .;*r- l=x {5 JSrr, OJ'a dr-r*r F crirL. ,lt '?t V vS u
! c^..1 ,ala$il +laii.lr .,li ,i.:r 1 r!,1 ..i:.:.. .J,l2lty--x Ja; ,g,1rVf@=x'+iy'
f '(x+ix)=P
;L*on Cl-f .0.fe3t g.l:.1 J1l ,;r' jf .rti:.:, rsl.llr o6y' -r:,. b;L--r
CtJ b u(x,y) c-,;- gb
u otJSi *2,L.2 +ti;tr rrtJ,-t ,f.Uf(z)=u*iv o6y'..r:Si.r-r.r,)L,i rJrLr;); o)y
I :x.{ \:*i .it'j u.lly=-vx ttlx=Vy ;t Lg.-g.i* t il, 2) .J::....1 .,t --,o 4|5; v 5
.vo*vo=c J;t:.$ 5uo*uo=. foL_./ JeV 6-1ls.lrr5 F tu ++tr" .-t.il,t ,rlJ*t ,*Vf(z)=u*iv o6y' .g.bu .;l,..".^ gti;1,.".rr glsyl2v
ru jlr....l ettt' tv isl j g JtV 1 dr-*l .,r5cr)rL, q qi t{ .l*i -r-!U c.-,,t .2r
6pota"r ,s:*lrv Jl- Its*tto=Y-Y=o o63in -x''y'o6y' i, .ri o'at^rl)vr=-Ur=ly
f*rlr Ji;l 6r$tFt t*,v=rry+h(y)
lx+h' (y) - Yr
v(x,y)=lU
f(z)=x'-yt *i1ry=7r
.C*t-rf vy=lx+h'(il svy=ux=Yx ef qi q
p*lr.f orr51$;l j'e At Elth(y) V $h'(y)= c a+# J) J
t+i;lr
lr,F
(z+Lz)-f(z) lz+Lz l'- Ir I'Lz Lz
rrAr+ vyAy+(Ar)r+(AL[Ar+iAy
tim fQ+Lz)-fQ) =
_ (x+Lr)t *(y+Ly)Y-xt-yt __
Ar+iAy
M+"
limM*"
limAr* o
limAy*.
lxLx+(L4t = ',yAr -t*
YvAy+(Av)r =_ti! J
iLv
.J.::.-r llt t'\z*. ,Jljl q iy J> 3)
0u| -=
roy t (x+y)
.e,,,1 ./J-t "#tt 5
cJ..l ,y Jit-, .?.Y
.F\r;z* o ;) ;:.:, s.l!" J55
Lz
f(M)-f(.) = r4,r, =d:;t=o'2) .':'rr&,t*Js*'qJrsl r"
Lz Lz LzA^;e.r) .,/b Cb c-l U-T .+t r t6t o22.e 9l2fQ)=(\ +i) (x+y)'
Ctt .;:.:, .Y
f,:,-,,1 dtJ^t 1"l:!.,
y = (x+l)t e u= (x*y)' .',r'r rt(;i f(z) :u *iv g_-^X o6y'
rr**'*,),t" oy ut
0v'
AJc
cld (- ''*- Lgt)ta, Cb ,?J=-x t *i.il jl(x+y)=-1(x+y) 6"i -5.-,
;l .:-,1 JtVy=-x L; cgst J 6b ;:.:, .c...=; dtJ-t f" ,;t+ 1 c^.,,1
f (x-ix) = Y( \ +i) (x+Y) E9
v=-x
Ar+i{yil.r rt .c--+
f_rlr Ay=At L; 62t e -h ttI A'lAr+iAr
Itrvr*urvr="
Itrvr*ltrvr= "
..r-l-rr J: SC i.t-." p-t.rSrJ:L, ))z=. i!,4 -trf\)='try Cb.5-r+r ilLs.l
.c-J j* 6!b rLjil
z=o tLfrJ)v= o ttt=r/$ f*llr
0u I = fim u(Lx,.)-u(.,") = Iim .3.Ia =odx I (.,.) Ar; o Ar Ax* o Ar
,d: b;dr-5.-ru rlly.r, irii"-r. jg
= *. 3v I
' ar ay 'ui'r)'tI (crc)
= o;JL.:.*^
i* cgbl: c.rLa:;- .lJ-r.1, db U-l J:..:,-,1 i.pt c,,t-a:;., .lJ-, i I F-s ,p
tim f(M)-'f(") =fim[,7+o Lz ff;;
rim \4]Ar-ITDI =fim
Lra-r. Ax+iAr Ar* o
,. lArl -\LL'
""tr -.- Ax+iAx \ +i ' ,#*..#h*.c.-.+ tf-yf (.) ;-ll\
.t<iT -t^itt ulJ,..t f(z)=u+iv o6r.r .f
C*l*f uy='v, 3 lta=vy PJ F.r)rL, t t+i ti
t- -uxvx=ttyvy
( rrr ) lr Jn
ttrvr*tt vr:. t_ -uxvx=ttyvy
.+l'd c..,,1 J:.r cgblr ^trbti, gl2fQ)=cosx*i(cos*cosy) gb ;:.:s .0
v=cog*cosy , u=coJfr,
ur='slr7x
vy=-sinY
, uy-.
, Vt='Si'nX,
r5c,*,,1 llll ups tp Cr-_l ,rSoYrL,sittx=. t shy=sirtx
x=ft:tt t y-X=ktt
btU; >tiJ.tl.ii;.r t{r ii Cb ubtq
y-x=kn ; k=o, t \r..., x=tant ; n=., t\...
;l r:.-l cr.lt , Jbl;J dr:i? ;r rli .;:.l.,1 ra-,,| ,rJ gbl:
f I nn + i(n + k) n ] = -sinrut-isinnn = .
lLi. cri. . 1 rrr )
.)LiJd.V.f.Sbf , U_ I J::....r ..;:5 .Jbb n gli;l ,-i*rn rf|, .rtti, j* ,-,irrr'.; \ +i tt - \
.J;5 oat^I J';."+ #-l.r
a) f(z)--zlzl' b) f(z)=Rez*Inu
e;x-, &:.-^ dlJ..t * \.1 "l=lr-'.-/-i,,i:.I
n gli;l ci; db dtt.i; - Y
.r;fb) f(z)=ilzl'
d) f(z)=(\ +i) (x-y)'
fl f(z)=Atv
+k f., tdi jL*.^ ett' Cl-f rury;t --.o J.p,e2: f J::.-.1jt.. ,r, * gli t-i - f.*g.z jl .r^rb ;:y,e I b t+i t{ rbt - ,ru^r db r
a) f(z) =7t
c) f(z)- !"INE
e)f(z)=+
a) u=dcoV
d) u-x'-y' -Y.v+gr
b) u=(x'!')'
e)u=r+
c) u--x'-yry'
fl u=try
.+t l b t+i;L'-.rr e1tf.l sitr .11.."*r n gli;l &^,r^ .s.r*t'.r cst* ltb sa - |a) u=e*cos by b)u=cos ax cosh by
.c--t' *rt'D.t<ji lt-f l<\t- lzl<\ r5-t.lb ,J_Vz {r.r aeJrr+rD.6re-0
,gl,L+ hVt.;l .r(.,.-r. -f ,6-.r .r;".bD &'a...r*.lUt &A..51 \2. g;;- 2
.r:-,,1 oi .-r*^lul &ij .5- ..r*..r j ju lr-p,ar ,Sr- lrLir;l .i*r. - -iJl
.-uiL ,r*-lUt .L.E J.[j ;rllir-/.rrb. oa,e.'t.-i- - ,.1
lr.F
.g-,1 oa5t U:i'- a^;e 7V gfQ)=lz I' gb- V
ef c"-,1 )3> 7tz.jl F-t--^ 'S'- ,t(fi 112.) * . J c)Ji &2-.a. lf|) 6b .6re - A
f(z)*o ,r(,r1.,"v, ;r2l llz;a cSl;l q
Jt c^-.,t .+ti Cb S.,J.t<iT rit .,Ltr D -.;.fQ) o6-r,o - I.$U JA*D t;;7G) - -iJl
..r.!U c4UD Ir l/l - ?.ri! u*uf tl$'D )lz -p dlil q - C
.J'-lb.iti L5:Jt Ref(z) - >
..:,,,1 c^:U.frbf(l rl(;if @=o oJlJ-.r to)J.ru.^tD tf|) 06y'- \o
a)f(z)= lr I +ilyl,
c) f(z) = lRez)\ -i (Imz)'
e) f(z) =cos(x*!) +icos(x-y)
g fQ) = lr-yl + i ly+x I
+5lr.rtLt.l -t*t* J::.-e j:-:., gblt S,_PW -rr b n 6li;l &-.f j* - \\.t+S
"a;* *:.-..r r,tJ^t
b)f(z)=ft
d) f(z)= ly I -t Ir I
f) f(z)=cosx*icosY
h) f(z) = lr'-y' | + vt lryl
d--J.+.o,Jt+trt* e ufL"ri" Cl-r ,iti.fuJL^ Ctr.l
&, Caf lrw=f(z)=u*iv 2z t *i-c1r 061 .eb.f ;tr Ctt.(1 ttf|):t Cb
.r.l* t, J<"r" t otr.lb.*b db if-l rf .:*,,1 c+-+.v--y ttt=x 1**rb .Kii
1,,..1$
.,1,-r+ d/-f tt u:li-.l jt-'u-&'-rJ ,rc-* 6-tlt Cb g''l .lry dr-Ja^ .JS6' Ji-. -#.t-.-Jg J*..#.1.1*
lrli. eti . l trl )
w=z*b Ctt .lt leL;, 6;e;.r .zlry &- otf t'. tsliiu.lrz=x*iy LLi,., )& y' {<r_l I -bEl: l"b, rJr 9.r Lfr er;ri tF &il ,3t{Jl 'lzz 5C/1,..€:i, .h:, 6VT &a
.tlli C-,rr 11 b2lr; g.+ rJrti t:), 6lolo6 ;l ..rl;..
.JSL.. Jf=.D r;l.r.il \ E"is.rr* b ,F.* c-,,1 lb.,frb 6w=z*b Oll\
,rl; Gtj.rtV Ctt 6- {, !:.-^ ,:rU lcb,., rl.rel b la .li .2, t6w=az*b ;tlt- q Cb.r^
..:-,1 P Ctr.51 w=iz+l +i=-/* l +i(r+ \) L r:..,1 lrfl2z ,o-4 ;l ;^;-J^ ,!ry gr-1.(..f e, 6* ,rJbe'.,.:-,,1 lb.ff b ,!*gblJ-lU z +>)-,c )t ,6# t*yt=at .6y' L r- ,-}1*ir, {r*i., 2> 6lgt^2, lJli r-l
fr.\r5.|=xt.lrl:-rlj ! 5v=x*\ su=-y+ 1 r3-l t+j tl .Ki
f )du-\='(v-\)t rJtt^, r, drL, fr ul-l dr-x .ii,- U
.rt-f i Ab cgWU q F-, to.i4 +, r.*i..o 6_;l Clj t- tr,J:-!. ."lr--rl:p-g;r r grS Jb.rt lrw=az 6b .6rr irt, ....-,,f :Jq
w=pgio ,z=refr , a=r.{.
{,=;::'
(ttv).... lrF
w=pdo - 77.f0+0.)
C*tr..r 't<ti
t+i.,1l-r
0=0+0. t p=n.
f"L+lr:*;,.; F.I,-r* .5-.t:f uw=az*b .-t r+ ,w uz;l.l-t .,, 6lt'.;t^'-
j c-,,1 lj)b r;l-r.il I lEst t# sorya c;lJ;l t;b-r: :lol o;l-sl q ur't+l t{
J-r+ ,Fj*r.t ,-, r-t:S; ,-f -f b .-llj JUlt r .ll..r.lr ,.rrFl r-b[*il .16
u*r-J-.r )-lJr t) 6ll=.Zi l jt+ Clj c;L-l t;-tt i .* d-^, jl J+ ..:,.,'l .-1-,-.l-'r
.tt-{,; dbl J,i;: l=^ai gr-l .,',-;b glY o:;a
z .2tw=f(z).t<ri,fl (z)+o J o)t s)-Lrsz rtn;.;: w=f(z)=u*iv o6y' .f 4;i
u.S,-f-flt.lic-6.7b-o;l.sl rS;i.114 lrz urlrl,.r.r-ll; -P,-.,1-.:,-.,1 u{!.6d^'tt.JS6.,Pw t^i-.o
.t*i r:,\z(t)=x(t)+iy(t); a<t<P rf-1,; Ic ;:':"6-f :gtriIw(t) =f(z(t)) =u(x(t) , y(t)) +iv(x(t) , y(t)) ; a<t=B
!li. Cfi. ( lr^ )
eJrL, $.C jtz., J dV b.C,*,IUV +,rrir,,e 2>C U.a.t otft,t' 6-r-ll\r,'l.l_L.J
w'(t)--f (z(t))z'(t) eJrL, +c' J grL- tr dJT Jt.-.l r:-,,1 z'(t)=x,(t)+iy,(t)
f-rlr 6rLj orl q +i tr.tr J^biargw' (t) =ad Q 0 ) +aryz' (t)
oL{;iV--sd @ t ol-tc tc' 6ta5:,. J. ,-,,W \; u3l; ?F:idt u.0tQ.6y'
0=Q_rltf-llr
6=4t+0
C' ,, tC' , Plr dd ^t;7l)Q" try le'.i,a;r gl-lcr jc1 ,;t>,-t JJ dr:l t,j\O.Jf UU
f-rlr ij qp q qj q.l*i ,yib.uv t^i4J)cv 3c, 6l,r.r:* ,$l0.=0 r-0,= (Q rrl) - @,4t ) =Q 14,=Q.
t-
.btd.lt-t, q f(,> crt tlr
.r,sl:A-t r.,-l+ Clj c;L:-i \ s gf nJ.eli ,-/)tq fi.r+, Jl-
w=z'ctt.f
j* aSz= o thi, )+ u. &it g,) I rll ,rIJ,-t t+ r-.r 2)w=z\ --x''l' + iYry Ctt
Q-10 tp=rr ,.*t-f w=psif 5z=rde Fj U.c*,,1 u,-r-JAA d^rtt *-'lp t(
.:*.il: f+fr 0 o;l-ul q.ll:.1: &--lr o-;lJ^il t .r"t+lt- Jrt-i|,5- g-lt* .l-l.lr 'f-A'OB' = 1 oo r,,-lli q Jj ,F 2;AOB=f 0' rr;1., +5 r.lr., 6;; Jli ,; JtV q I
.t-*,g ,hJ'ju, r-ri.r;r
.l-l* c..it<i U-l\z=\ tz=. Jr[.At .c.--J u-r-J.g.:rt-.a:;", .l+, s)w=Zr ,-.,uy-
lztn d,Jl c.a.? Or* -r.Sf J4 .lL:r f \b l"ti U-lw rf.-t cr.rlt q t- $L./ -#
p .4,g c-.') ^'.f(z)=z d.:L, J- ;t w=f(z) .J*'+ crti JaUli .rl5 cJt-1r
.Ji:-,rt.Jr-f* v=Y ttt=\ -frL:. c..'.1., b jr *y=+ t x'-y'= \ .trL:. Jfl.i"
JrLi, Cli. ( ly. )
-m-Et
j,t il
t!/-;::LI
V-D/ rg
.r2
n=l6b.Orr b .iL-.d;., .l+., ,-rlt ! uJV -f1 rL-,| sLL; z=o r.t 8 j+i l"tir 4-.-a J) Gb y-l
dby.; 6Lr1l.r,;.: f 6r*i ,,fr.sr"/ -#l;i *a, J: SC.J"=. r;i c;lJ;l
..:-,1 o.Ll o.rl.r
( rYl trF
t-r=* .Kji w-peif ,z=reie f.rr ,t} ,6g.r:.-.r j*r+ cr-l c+ti !ti;z=1\
.:_2r; o:[i:.,,| 6h:,-T 3 gg,,loyl>.Jl.1.,,tfu I 1l ;tl j 4w \z;l or.-, t1;lr4Q=-0 s Pr= \
.r-<s l)OA t g\B iaiEo fy *, dtf oyl.r,r c9ljl q S €C,S.rlirL-
1l.jR ,ii .p 6OAxOB=R' 06y'.r:.bR 1Uj sO f-t r,6loyl: u.,.>.;A tLi'i
'p_.:-,1 ;UL, oy-l: at.euz o,Sstz\ €,?t o6y' ii.)-ry;).rll oy-l:
,J*rt, l.w u.ie -ssz tLi; ":.4
;ll L; u.uz..r-<d".w otil;i ''tl lrrl =rp= \
6;loyl: u/t<'il 4,.if ,'.1 tu=+ J,-+ t o.i., gz ":i,;,irllq
oy relp *=L
c-lt<J U-l .i) C \.L,) q ;rl )e; $ g-i sglcr-i .,"K^ll &-: J-l.l oylr r, &*i
;U; oylr gj q l.r .r-lr GVt oyl: F J J-lr oylr ,.f-l) q b l-lr oylr E7t-
.Jf6r.-lr-.^+..r-<.Jt3 c*.,1, L; uU- oylr 5-- q ! oyl.l -l;*=+.!,--^+ .J-S6.rl,-r",
J-p,e u z t->i.,c.;r o,-,1., Jai .5- b- cylr ,S-- 'ts
iJ:t^,
a(x' +y') +bx+q+d= o
\ - u-tv--.----------u*iv u'+vl
-tb,i;r ,r=* c.-,,i JliCn=+ ! +i U ..:,.,1
LL;,Cf-r. . (tVV)
r+:r.,rr Ju
rt'+vtv
' tt'+rt'
, rtta(-' (u' *v')'
p*t-f jy rJrLr;r crl-rtf ;rl .lrrS;f;db tr
t-.1
o *bu-cv +d(ut *vt ) = o
n=!.J,-r+ d/-lJtl
I tV .r"i, or4ls.i- {r. ! crL.as", .U", l.tV ,"i> oy'o -p d*o ta*" 6l;l q.*.,iJl
.J66.,-}1a* .iLas- [ !,,f,-r+.11,. J lV,"b c,-'lt !i .5.r- q .hf I lV oyb r.o d=o to*o gl;l q.V
'Jso'.J56..,[1+ .lr, t tV oyl.r r, b .l-r* I lV ,! c-,-,,1, L; gd* o 5(r=o dljl q .C
,J*+ .fJf I tV c.-,1, Ir 5y t b .l+, I tV c".'b L; gd=o ta=o 6,1)l u..a
-t L:. J-lSs q:+j i.'.y= \ JI=\ r,Lr brA;.J*ry d/-l t{ .JSrr,
u'+(v. + )'= + r (, + ), +v,= +
rl.r'-l-./ d-rt'
,tx.y"[ry \rs.# F "kryt{.rB.l'.-[:t-i '?
\ e -J ,F ,Jbt-1, "h-ri
t,ad-bc*' .li 1.r .5w --H c:st' r., c-'163.r,o
.ttj .l-,r q.i\ry t-V 4t ;l Jl;u. t, J-ry grl ..:-,1 lyy d-H,f ,J'ar",
Cl'r 'rt
,=+#=+rrffi)=+0+W)Jtt' t. tsl;; u, t, ,:T .S
w:A* Bz*C
oylr tu !:*i g;loyl: Lrrl5.jl &-lC o;l-ul qz JUJI 4,-il-t^r C*lr:tf .rJr.'")t;
,f,-ry ,rb "f,-r*
,5- +1 dyy ,J*r+ .;ll !i { + Lttar-i u/t<-il 4 r 4,,>.;i.,o o:.-hv u-,-r: r .riJqp Jl;u.w=#J-r* tsst)l.r-r,l
w 1^i-a,.-r-b"; thaz 4^i4 ;lz=- L ,6r:.Ai .rl-,,1 4 q 4 cJ6.J i,1l! w tZ
17 ttltb r-.o-1,.J*{ grl .rz,-,,12 A^;e,*r-t+.i &iJg$w o-i-a )l+ ^bz" td \ ..r r. tl;l ..^A .!,',iii ,- . - *'=, @t . : .- ^l *.,J-.o ,=- L)34y2=- t tt )? $.laLo +"-r.1.r d 6z:.w'= ffi ly; c-.,t *"*t=::
I-7 \_
!hi.cfi. ... (tvr)
/-r* ffi.i- t{; r5c,*,1 gr-l .: --..a1-- rli; c^-,1 ;1." c -^.-al- dbl) ,J*-t+ Crrl
jlw, Jwr Jw, tLit u- 6tt 1'.4J u.lrz ^^;e ;lz" 52, Jzt tLii. u.,r5.:*,1tff-f
.r.r.li r1.r:r u.J;6.lb tgU il JliC t,.!-r+ U-l st.rK./ w te.i-.a
W'Wt WfWr _ Z-Zt ZfZrW-Wr WfWt Z-Zr ZfZl
+r-+.-\+ Lvt,sstr..*.rtb-\ r'r \ l,tii,,= #.1,-,+y,o t-\ -l* bjoGstt.rritb-l J. J\ t'tii *-- *J.,-r+dr:it.l.r.rK.-/
.t.lK.r.
ueVLS.# F.)!,ry.Ve-f 6s: +'.ls!>. j5l ^^;s t+ Sp*t=: AIS-J ,bt.J*.^", €lf ,lc,.-,1-,, b t- j oyl: q l-l .>b y; S-4 ,b,-}1r1 .5r- 1(,:,-l u.),;.rtt.(.1 lw I s t
b J-+ ,-.r:l .r: ,i1lJt+ ,_rf J+ dt 1;y>o r..r-b df+lf a, V.l rS..f J*
Jl Ur.-t.ib I , | =\ or,,lr .t U oy-l.r u-l s >-*.J,-+ cyl: .r, 6.:*,1 ;j).),=. o-,1.,
+j tf 'l.li-ii
g.!,-+ lr l < \ ,f t Fl, rr Cl.r 6loyl: q by=. l"t J-r., ,fr6;e ..,' Jr €lt sy=. Li q &-)t ltii ,:,it ,S-; Fj*+ ,t ,PX !
"$ r:*,-\C *W ,* j-hjl -r;-r-, €l: et Jtlr'1r .:*,-U gz +z"i-o;Ur.;
uAV 6F 'b gb o:; 4, t 4 ! Lry nrlr r:.iL jVt t2.i4 d jl Jbli)
,J,- Jt--.S.-1,-r+w= \ oy-l.r r, lsf =n t; -t-! ,JF ,F "1,-r+
ffi d/-llttt.,.:,,-,,1
6 €e w st*rr rtr-r+ u-l: f-$,1 -h -ltw= # oy,.e t', u!;.J,,*r,.S.1iylry Clf .-rt[ 6il u.ltZ=@ tZ=\ t Z=o tLii l-' jg c- -,--aVy. o1)tr
ldl = la I p*trf .:,-,,lCl: I wl =\ oy-brrz=@ tz--" loUj ul.i; r(,!-t q Jb t-rQ
lf,t=r
dc
+({l u.+ j\c*. sa* " F.\dad-bc*. !qi\ }"h;l lc I = lo I -r
0.-f \ sw=+ #;l orLn:"'l u Ib
a
1yt-gia Z'Z'
Z'Z t
=lz. lp*U_6 I
lr,F
,L:dotc
t'*t-r
Jz:\ .Li ufi: i,sr-' I z, I
l\-r,l=lt-t.lf-"rlr ri-,,l qill d.ral:
b-3
(l- z1(t- ZS = (\ -2.)(\-7.)
Rzr--ps. ,y-l4\zr*7r=2.+7. 6.\'127,=zZ. Lr I z, I = I ,. I q qt t1
u*r-.L! *:c4U,j,-J{ l).zt=z.Jlzr=z.U-zt=2. g:lr I z, | = | z. l;l r
lrz. Jl Jr-..r;u n=* *l*r,-t.:Ja r.u,.J,-{ drltq )-r,l..J*+ *=do
r^ .lbutl d,'" t o"p -rr -l I z- z. I < lz'7. | .6"i C,,"d i\t '^;e g11 J)
q by>. j6i .,^;e d jlZ. ,-,lill ,^ tr ; a ,-..,ti:jl ," ! it ot:n t .,J*Jl
.r-rK,_/ lrls \ gey tstt
il = t*t^3-r q
!Li,cri. .. lrvr)
6\.c,-J,ti l, | =\,rt 6rt u.tr.<0s * +V jljerdr JU, Jl2* +;'' r"sfl<tt r=21; 6ss \lt ltl +.2b wt=zr,J*l"i b c^;.: r:*,,1 JLt 2.JL.r 4l
-il .)$-ry jl ts ,tt_p,-l A+ drit I r:r o:te:-,| 19y _,t--r+ jl e1 .:.:t(;
.J-iU, -Y-
W=e* L 'L'
z'-2. , Im(2.)> "
w=d.r-lt* ,rt
f*t-"f .[sl J-rt u. +j I 5w=dP €C *,L,rs d,:i? l-2w=d = {*ry eb,3 l-o dr-r--l .r^l5 .l) r L, .1.r v = d siny 9 u = d cory rf c*, I,_rc- + w = d ( cosy + isiny )
w'=d ;r-.,-.,-dl tlt Cb rF ! .li ;:.-, J o).1,1 lb .*b w=d Cb Cf_bh .J;Ss.
d/fJl+ t+'M =d ^(!-l \ .h / * . f_llr o;l-p.r ly; c,..,t ,;.,_J*.r t7 r-.r Gb ,rl6-hV oK;i-n<Imzsn €,ito6y'1.r:-.-,1 lkrtio.rlr ! y.lLr. 4.Vw={ eU.t<ii w-,osif:deV
--,.-f.otf.r^ ..:..r1 jttw t-'.i-.e.1 r:rb dr,,_l
JrLii drd 4, q 4Q=d uel* L; u!=d Lt,p=ec ollt u.x=c ; -n<ysn h;
'F-Q=y :p=d
.r-rc1.},-tt,
reb qrs o r->b ,+ <O< | :ecn(er r->b ".+ <y< f , t <r< y ..-r;
.5:-',1 J-+p> \ ..3,b q.r> o l>b, " SpS I
w =situ cosiy + cosx siniy
siniy=isin1r, r cosiy=soshy
sinz = sinxcoshy + icossinhy
v=cosx sinhy , u=sinx coshy
w=sinz c-rKi.1
€C -L-ys C,:r? Itw=situ=sin(x*ty) eV
6r.5.::ti .lly.r,
r.$-1.r I
cl-r
! -b9 r:.-,1 .JJ^t t+ ..^ .J*+ Cr-l Otlf Q .J"So. oLa ,?-t.r5.r):Lr.;:+(,!-l
cosz = cosx coshy-isinx sinhy
f"-rlr(siru)' = cosz
JliCL.:-,,f j-ttl"L;, Clj )t ,f -*Ctr rr4i- Cl.;; ,flf;l cStt-r
.)J,.t C.1U
sin(z r*zr)=sittz rcoszr*cosz rsittz, i Z,zr et. {-dbstt'Iz= T ,
ly; .:..,"1 .:l-f gl.ll: l4sittz= 0 d.rL, )tls iru = s ituc o s hy + ic o vs inhy = 6
$r"/ {+r:1
Jal;i,Cl5i. "" (tV'r)
sinhy= " Jl cosx sinhy= " , sinx coshy=6
cosx= " €,-f ,f -fJtltq..l,,l 6- * S tf u.situ=O t"JT il s!= o 4 :4':t
* S ,:-,u. -coshy=O q r,,rx=ll<rt- ! :lr Jsr=lkrt+ ! 4ra;l-llr d
:l-* rsl.rlr tSe'v-\.d+\=. r/ d +v=llut* i :' ji '>-''t;f*yl-f rsl.rlr siru=6 e'l.:L, d/-tJt.t *i!.gl=Ln(6tl-h)
z=llut + +iLn(6+l'l?)
z t^ie J-nt<)c<n u>V dr:1 d(,- t 4, 6rb\sin(z+llot)=siru 13-l ! -b.o,,,|
.rrl.lr (siru)'=cosz.li J:.:, frr; c^.-=t ,s*.1J,al,z=x. L Job;:rl ,J*r+ gal .:,-,'l :l-rtl
Jrf.i" q U--l v:cosc sintry ,u=sinc coslry t o(J=c a + b.c,,-,,1j4 blil
-4-- "= =\sin'c cos'c
&q t-.1 v=cosx sinhd ,u=sittx coshd b.y=d> . L; 1
tt' - '' : r
cosh'd ' sinlfd t
.r_;,1.!r+
ltY
( \vl ).... lr'F
.-!,-r+, .^;aph CrJ) ggl+hii t+ 4=d<n \ -PVB ,v4 sA' +,A thh
v=o hi ft' q t=- l Jor ru=coshl2\ 1v=o b) d qt= i Ut.r:-,,;
l-\ Sn=sinrs \ Jv= . LiolL,qI=. jl - 4 o= + J";; ,u='coshy<-\.JJYY
.:-*,;,-[-^* Il=o L] e'lx=o hi
-T2
.q ,72-l..l.z = w--sittz J_ri €,9 oroLl ^+lr?tLi;-1:
>[
JI
Y
E2
+
LA.#t<itL.\.w=cosz d,Jt* L ,r3i i-.,r y .&t cr\q ,sss il Jli c b g1,.r+
-rl cs-rL-:
C.t2j,o +. tll5; u.12
w=cosz=sin(z* * ",
e J €r,l Jtititlz o;l-ulq b ii Jslcosz c-lKj \z '* dtrl{ Slt ,P-t>iy
fl crtsq ii 6lr4lt'r:+,}'r+,,:--:,y ffi Jli,g lrw =laru,J*r+ )Jh+o
siru (t'-r'') (e'b- \)w=- =:-:---cosz i(eo +e'o) i(e'E + \)*>)t, Jli r-l w t=era,-i I \
Ja.Ei, eli. (lA. )
. 19.- \w=-lwr* \
.r-*r./ r1.,-.* d-Hy c.-,rt<j 6- q .,!L., c-it<i &-;l J'.'.taru {-:,1<i 'p-
u. 'r:+
&)t1-t; 6-u tl ;tliC i lt JJi" d-# J ,g-t+-,.i)L_r;J;4j o. dr:t l, w = coslaiLr rrlT r.lg.r
w=cosiz=sin(iz*
q!..r*;,- t + o.,11-ul f JLi:rf t + oll-ul u.z lsl2et4,;l l^1coslu)t+d/lJq
&i-3 Itw=sinla.1,-+, J.}::lr .t-i\.t' J'rL- A.ra u-gtt g.lt(; 'jry.,t5
.eJd
ollril uz Jl2gsjl Jrw=sinlu,lr+ ,f-l )l.n),5 r.ag: ,- + oll-ul \ ;ls5;,.Yr
JT
,l
w=sinla=-isiniz
.7 lttr s rsl t ,-r--*.!r+ lflI ;
w=hu 6b .\ \(| t-tt o6;r .l:.V u.z .l;.l-rK Ctt b .li; -fr-r3 ,-f Ew=bu ,>25.,o t' l1e* =z gE
f*[.f .t<iT w=u*iv , z=reie ; - n30sn
,u*iv:rudu =reio
&l,4ti v=0+1fu tu=lnrleu=r t+i;l.llnz=lnr*i(0+ lkrt) ; -n<0<n
c'fif q Js .--'l 6.2lri, .*,-l+u 1 c^-ai Ctr 6- frK €c oJlL r,. 6^+t-,a
tt u.a;,.-.t,Sk &.cSl;l q d.*b..1-,,1 Ctr.{Ft* ol J:.t Lp;^^l ,Sk -fJLel r;L,t, l_2k=. q rbU:. sU 1 r:.-,,1 lyy frK lle iiU + n)e €-rr
d/fJq J..u,,rr d.qL.J Lru u.r J-b"/ frK
( \^\
Lru=Inr+i0 ; -n<O<n
trts
tL-, d)y..f."t.-f +-.lrl l) Jlal gt, t r--,-lt$ Cl.1i )V i C-^, L
tf 6 ,Ssl J.Lm,-r^-r-;t3 6b .c.."1 ubl r;t-l -r;-iLl .r--'-.,,t* 4;t'ocL-
.t<ji r-:Lt,rs". ,J.tg- tF ,,=9J iltt. Jtt*c-,,| tLts*t * ,f.V P €f+(--lb
lim Lra=lnlz.l-inzlz.
,lim Lra=lnlz.l +inz [2.
dd-t, jlt_!q ;lz flJ o)r u3. * ,f.V ,p & tF 6 6Jt J.Lru p-lni i3lt;l + ,f .rl gl.rl, 'l-A gr-l ;) 1 o)y tll * l- J) gl-llt J5 .}:a z. tJ*." 4.
c5blr Lru gV t otf O.,t-c t?*s ,F.r!rL, 2sv=0 ,u=lnr ttii /L.;) .r.:.-,1
c--lb & tp FJ;q t2!l*, b;e ,SJt Jw=Lru g,l;d/lJtr..-'+ o*
.tF -St tbb"r, ,.,rK gb r.slr-L-.h;.-{ b ,P ea- sp ft'.g-.,lrrtJ^t
^,.;edr:r ,Ftw=Lru oK;i ,',-loJ:l & tF F J*.;r ! JoL;., biLe FSld 'Kii a<Asa+lz € 'a;
-n<|sn gl+ q 06y' ..-:,-,,1 .jJ..t 6lo-t-l oJrJ
rtF 0=a L; t* J* t J+t s st $l.1r 0=a uelJ Jo; '.+
6lctj.:,,L.a*, nll, L:,-l 1r 6 xriSu. rLiJ S.-;l 6;lr.-L- bta; drii? L".r.rr*..f
.J-.o:.rr ,f-L; Jt Jtte q b tfJ-r,r; .bVr-l6lcL- &,,iJ b 6,lahib ,* g\,1
Jr.Ei..eti. . (lAY)
(. <0Sln)0=. g;lr;t-, t;w=fi t w=h Cli .\ y
:lf.1r 4;1.;b r5.:i c^b uw=fi, oK;iz=re't l-o<0=n olsrt
w={r e
.e .0,- t--t/re' t w=t/re Y
Clj;l ;1-t5.,e J.r -r-,tr Ctr &_ ril;,.i.:-,|
w=-fi e , w=r/Fe .0
j oJ-..b w=fi, 6rb r;t-i .5,- br-L
fr-.lb w=tt+iu f:=r*X
r0r0v={r sin L , u=t/| cos -YY.i):L, iv tu t|=n g;lc[-, L* a,g,grJrUii f .it..o*, .lJ-, :l * th;tn.;r o6i;i
ol-Jy' .eb./ Ji JLel d.;L-l lrw--fie 't "t
Vr=
i( 2 +o)t-
,0t-Y
.0t-Y
tlr, 0 \-su7 -=--u0 , ur=YrYr
fil:1 .lljrL." ,?-.t ,*$ro, q' = \ u,YVrYr
ei0:1 :_sin ", 1-Y 1,,/r Yf (z) --s'io @r+iv) =s'io ( fi tot
\l{z-)-- ,
tl ,''' = J, ;'i =l,lr YVr
n ..:*,1 *lr;L-r+ .li.;rrS
.(J;5 ,ru)e;l:t w=G ,-/)Jtt i:liu.w=G' elt iy ..,-.l -r:;Lr
w=f ,gli Cb.\frJ:1 gi;h.e... .r:a c-le uw=e'ht .r-lfQ w=zt=e'oo €t '-t r; ffi l1w:zc
w=e'tu Cb .f-, J. 1r. tg, .:-.,1 jK "r--,-;t* Cli d-t., -lsLl .c--:/ 6V lru
i' ..-,L, q t.6s t-:,-l.1: t €,-l.rlj5t, orsl;i \\ >:-fil-l.lr d^*r l, 6..:;lrr,,
y-slt*l
-4@tlr) = g-nth -\
t/en
o-t-l ,y JiL-, .1.f
"q+ (r +4 r. r+ bw,sstr.r*rtb \ r' tiJ'Eir5qV ex-r.J*q'r
t .?j t1
W'Wt Wr-W, _Z-Zt ZfZrW-Wr WfW t Z'Zr ZfZ I
n- i "-*rr*,Y
"-+(r+4\
_*- T' T+Yi - z-i \
"- + (t+t1 Yi z-\ i
@- + )(t +r1Q- l:(w-{ r \ +i)) (z'i)
1i -rilni -ril:rc(nh)
z'i o- \= z'\ .-j
f,ib
.- \ iY
U-j
lrl:i,Cli. . (l^r)
w[(t+t)(z-\)/- + (r+4e-l = - + (t+i11(z-i) + w(z-|
w(iz-\)=- { f\ +t) (z-il+ i ti* t)(z-\)= - } {rr1=-,
;l .:*,1 .i.lt, -h tr-r, J-+ {.+;j J)w=-+
lz-\
g-rl.l r.rQf ,t s@ 6stq titq br, r ,* J,ub rf.pV.*ry.-r&rr,.r
?- ? ,-\*Z- f-* - ,+ E0y\Y ,-\
-
=*rrt0
-r(w- t , o- *,=tL e-Y)
t;w(2.*,=-* e.y).* e-it
z+y r.n"i -2r 1
w=J-Yz- I
w =, (t+i1z-1=(t+i)(x.r,,-rI,li.ro (t+t1'''
'J'''+ tr l'rv=- \ ld uiit 'r
.+Siy=- \ q qi l v=r*y , u=x-y-l C*t-.f .Kji w=n+iv 1.=*r;, o6r"
v=tt t-j rlry ta;i;l ,v=x-\ tu=t-\ ,-*,-l
.t-*C.,1,,-r* uv +*)-o 2)v=It b 1ry ''^;r )ly='\ lai ,,f-J*ld w=sinz c..!t<J q b \ <y<l 5'n41n u>V u:ii 'T
w = s irlz = s iruchy + ico vs hy
C*trf w=tt+iv ,ra;l'.
v=cosxshl , u=sinxthy
€'h4...............Y = \ jl
ttt , vtL-: t
cht \ s/rY \
,r..adY = Y jl.l
-4-*{= Ich'l sftrY
.t-*,1 Jev
ll:i. Cl-i. ... ( t,rl )
uv .'-ir.e t'Jt<x<tt .lt drl.;,[rb t sr!.#sittz -rj\ d-ru,{.2rr.l,.7
t tt= " Jai *x= c dr::',r--r J5; lt+;l bfr: -tt ptw, j-b.2lj .:-,1 6- q 6_
oti t-+J) dr:l {iL u. }rll.pr rsb J-r.. ,.r-,, \ <y< Y .lr t t-; r.lrl.,-+ v=sh!
.c,-,1 cJ-! rJ.rrr v 22r., Jfb.lt 6 t_*a,!r*.+l"t w=d.!t<i q b . <y< \ j . (x( \ q-tj,Jiir .O
- t .l(?trl
('*t-.f
z=x*iy , w=pdf
Q=y , P=d{+# i::F€.},-+ "<0< \ ..-ti1.g< \.,l_r ,\<pce r--b1o(r( \.llj
.,11 .:,-"1 g.lt r -h tr-t J.t* q2b
'-*rf ..J**rt p-, ,rr-l: f;. e., oy-l: &- f-tn=LJ,-+ ! .5r+, dL-:i .t
o ,l:a(x' +y') +bx+qt+d= o
.5.:.-,1 p) atrtf ri &,l oy_lr r, oy_l: .:-,1 .jt .(!-l u. ); .flCF.r, 1.,,
a(u' +vt ) *bu-cvt-d = o
oy_l: r,, jy oy_lr jy,)_+ \d*o ga*o
4i,l ffi b Y-l-l'g"-l 'rY:L' ';,-r-,-/,J*'+
$*+)'+ o*+)'=#.*-+
@*+)'+ o-+r=#.#'t
ea=dt+i;t t+ = ++5c',1t J tr,j1r+,&rr.,-rr oy-lr st fl-t'<'-tglrrdtii"'r
c =-cta ld4+#Jr.s-,1 d*o sa*. ,i-f;>u; tSa=d=' r'kl 3;)r+p fa=-d AV,Vs
.t;s1},.r*.&r* q b ii J-tst ftY"= +jr'+5-t+, dts.V
lsin hyl s lsitul scoshY
f"-rb
siru = s itu co shy * ico sxs inhy
l riro l' = cos' x sinht y + sin' x cosh' y = (t -sln tr)sln ht y + s*t x cosh' y
I ti* l' = sinh' y+sin rx
=s inlt' Y
Ititul>lsinhyl
I ti* l' = cost x sinh' y * ( \'cos' x) cosht y = gssh' y -cost x s co sh' y
Iti*l scoshy
q.3;r
e .hil
d/-lJtl
f*rt^. .:-rl .-r,Jt .r\
t r+ vi -, (r + il)hr - r(r + i) (h r * r luti) - rrh t - ilut + i ( r hr + rkz )
= {cos(lLnl) + isin(fLfi)l ,rtor'tktt
tanh-'z= I 7n \*zY \-z
- ew-e-wz:tanltw:fu
\ *z - -l{_ -_ etw
\-z Ye'n
H/=+ Ln \,*, bJY \-z
d) Y' =xt+ \
c) "<x<\rcfl(\
d) lz-Yil =r
r5-r".lr dLr.q
t+i;f ,w=tanh-tz l=S ,rtf
L-1
,itlJlr
\w=Ln \ *z\-z
,Jlfic)-71argz<-
.)qJ*t .\ o.f.+k w=z' cJtJS tr tt y-; a;$*:...il .5*t .FJ .l
c) y-- \ +x b) x=Y a) Y='x
:+:td w=z' .JKq b tj ,.flj;l &*r" uiir.yb) "sys\ a) lrl>V
:+tt *= + c-iKr ! rr yj 6ta;*,;t &; uiir.fU-+.o,w.? e.<argz.+
.+k " = + -L!Kr \ r.,,y; ,--.liil 4, Jr .r:i .fc)x=\ b) Y = r - \ a)lt+\l=\:+k a) Lgla:-tK;l .5r.r q b . saryzsL r.ab u,:ir .O
e) w--zr d) w=-iz' c) w=iz' b) w=z' a) w=iz
r5+t: !r-X-f ..1,_"i.f
.trQ+r+e \ rs.r.r -r\l r \ eo thfiu"'a
.rl&€J-\J. cil..r lUit" t-Lb
.J-i.U .li call &;t ,S7z= o.c
.ty3{:.t,t<r w=. t1t) Jr=+r5r.,,t(:1 Ct* l,u I s \ d.l.r J b I zl s\ +.-b-d
.r.rQ lwl=t &.,rt Gst\l2c1arp.+ u>E-e
.-t.l.iu ,.ri c..tl lrti;t t-i bli3.f
.r.rK./v) \ 1^i4 6 tS-ttq br> o .>b 6w=iz*i .J*+.5-t+, dL5 .V
.J"rtd w=(\-i)z,Jsr.r ! l-ry> \ ..-b !i,ir .l\
.+k ru= * ,j,--rr, tr bx'-y'= \ JJi.l #;r .qz
tr ,-f_-r* Ji .t<JT J-iq up;9t.Jr-.^+ 4, -U +Liii crt a:;., .U.:, Jt .Sr$c4tj .\ .
.<.fcY ,
-f <r<-Y ,
flc ,
.""''j ,=fr Jtt' Yttlr ,;c-rt<t Rews c tab 6 /lst + lz-t I s t ,-f -f "=* J--^+ tr .5.r.r, dtJ .\ \
;-*,9.r-ttd w=d,,[1r* ! l.r yj ,fli.il .i*,l ,:it.\ Y
-l<r<\ , -t.r.it". <y<\ (b
o(y( o ("tlr
. SyS ,t (d
-\SrS f , -n3y3n (e
.+t.t w=sita c-it<r ! Lyi ,flj;l &*rn d!+JA.\f
.oaL ,Y
.<Y< | (a
rrLA.cli. (lt. )
Jt JT--<r<-YY" <X<ltt
"<y<\ (b
\<y< Y (c
s|'.! 6u \&- 1 4.t* !lt(.<c< x=c L;w=sittz rS.rrrr JLr .\ f.r.163./ JJi..5- c.-,,1, €.,e.-,
ctt+Li..)U, . SyS \, . SrS ? *t )J. s2iiw=siru,l-.r+ c.-t 6.r+, iJL13 .\ O
..-:,.,,1 ,*.6-;l Lr;..-|E' D' ..ti.;t r5c.-,t8' D' ,locn tA' E't A' D'
lo; o.lb, tSSt ! lt ( " SyS ltY
) z=iy b ..rU, w=cosla cJt* .5.r+r JL5 .l t.:.1K./ v--. t. SUS \
.r-163./ v7 o L>.V 6u u.l2y3. , o SrS * .*U w=sin'z .!,.r* .f +t, dL:t . \ V.Y
tSu t.!z " t- ! s.rs f t* u4 ,rliiw=(situ)',t ,!-r* c..-r ^f
.rrn.l dL:r .\A
..r=5 oet -rlt ,ri,ltlf .l--.J ,l €.!Kj o*,,1 eltu=y la; tt 6 J-tl g-r jt jt*s.+td w=cosz)',+ tr l.r y; ,-flj.rl &*p
":n .\ q
SyS\ , .S.rSYrt a) !2" , oSxStt
,t,t
Dr+
.rdt=: w=Lnz.)-r" q b ys lz I sf .r I sOs f *t d:"ai.y.
( rr\ trF
,-#t,J j)\ rr-i J) rrw=z+* .l-* 6-r+: .l[-sz ,P Jv,e;l orlrc"'l t{ 'Y \
.r.16./ v- . t-l sus V Jo; ,2\ tS-tt q tt lz I = \ oyl.r
5 -t+r .JLr .Y Y
a. cosT--W , sittZ =Sifr
b. lsin hyl s lcoszl scoshY
c. lcoslal scoslu
d sin-'z=-iln(b+/( t -z' ) )
e. coE-' z=-iLn(z +r/ Qt - \ ) )
f. cosh- |
z = Ln (z *r/ (z' - \ ) )
s. tan-'r= I Ln \*zoY\-z.+t=t lrsiru=coshl d:L, 6trr-:r, 15.If
.+tl"tb cosz=Y drL, .5Lrr-:r, q5.Yf
.z=ntci ;l 6 : Jl@l=cos(E)'S'r+t dL5 'Yo
.+tt=t llsinla=i eJrL, .5Lrr-:r; 15.Yt
-St s.r;S.:,U .Yv
Lru'=lLru.Kii
.c-i j-tl'n<0<n tSl;l q g.lti Al J:.rr-,1 .rlJ-;,rs o , !=t Ja,* ,S-tt F.t- q-r WCb 6J--o: rJL:i .YA
.J:i-,J l"b, )Je 4, Ott' \l.t t;,.:-,bqt;l 'S-re'Yl
b. (\-i)"
.r=rt r w=(\-i)zj*r+ lr by> \ ^.rb ujij .f '\
.+t=: *=:,!r+ ! l.rt'-y'= \ Jri^ ":i.r\
z=reio .Yrt
, f)oY
a. ( \ +i)'
Jrbireri, ...... ..(trv)
.r:..,1,F.jr--r+,5- P &r+ 1.r,-l; .j.r+, OLli.fy
, x2 t t;4 J* t c.t^! rJ.r*y z t.*i..c rf+to*,r r, Ln(z-t) ;l 6lcLi .lr.r.rQ w tai.a.,) o (p( ln jy 6rJby=.
.+tirk il tStL$^:,63.*y lrw=cosla.t.,_r* .ffz=d , w=|e*)l
eak?@
.hf:*rCf-r :lS.&|ftJl
.hLi, .at ,o;r Jei 6.lt JlFl .\.fbc ua:n )jh;- Ul Cl-r:,..ru;lr;.r, ,F c;:t Jl.r(.sl *-.U+ q k, J"Li, Glj.r,
rJrL,
z(t)=v111+40 ; ast<b
cJ-l ,.iud ,F .;2[;'rfo:y t ,J44liw=f(z) €e ,,.t t t#,1 P 1olt
(.t"t {rU tjn u.a=t.11,(1, ... 1tn=fi /-rE :U'.sl \lt a<t=b drl,i .ri,U
e .JrJJ rP-rc 'F lsn,... , sp s. Jatii r6;.j ,1r.1..," Lt'y U_l I €rSclF5 t* u.tr-rsn , ... , sd, srsr .y cab6>tu 6il tlJzn, ... , zr , z, blro0J5l
n tfrt ':29.e i J-s-r- o63i €g./ti.sl
lim&=t
^u lMzl+"orlr rtLJ !, f(z) fu r>2;e *.5 r:*,1 lyyc uta,t $(z) Ctrb JfFf q
OtfJt t .t-#,tf" flr) n2 = tim :, feilM*
** l&,pl+"
. e.\s, f(z *) =u**iv * : f(z) =u * iv rs;T ; I U.
lim f fQdM* =lim f, [ @Ar*-v*Lyil+i(upLy1,+vk\xilJ&=t &-tnw lAz2l-o nw LrpAylr+o
d/-lJt{
bb1,fl4A = I @dx-vdy) +i! ludy+v&) = f {w' -uy' )dt+i Io{ut' *ra' )dt=
b
I"@*iv)(x' +iy')dt
rt-ljlb
I"f@at = ["fe@)tu(t)
o6;. it ,rlr, J. iJ.Lr,- Cli f ,.,L:Crt b ,sst JlFl €lril 1.9Jq-{
If@lar=l l@d'+l f@dzc ct C1
,.5.-rtt ;tli c JrL q .6l,ii c=c, *c,
w=z' Cb;l .\ Jt:,oB J*.l'l- '-'ill
.l,_r$JFol
OAB Jjr;r- .,
( t--toB: { .srs\
I Y="
;l c..,,1 crt\o OB 6-t'ft, rJrt^, . ry
f"-llr
i,evr.
dz=(\+qe , zt=(\+i)txt , z--x*ix=l(\ +i)
z,dz = { O+ifx'dx
OA: .S.rS \
dz=dt , zr =x,r , z=x
, z'=(\ +i)tdz=idy , z=\+ty
+[AB
IOB
!OAB
zrdzz'dz =lOA
( t=x
t ,=.
- (t+i)'r
z'dz
;l .rrlc2t e qi \AB sOA;x=\
; AB: { .sYs\ ,'
I v=v
eJulffiOAB JJtfst
tSFlt\.r)rL..
f-rlr OAt
g-tlsAB tr Jr::.r
d/-lJt i(t +i)'
=,&* rI^.^r'& = !' x'dx +i I' $+iyfay= + + + (t+iy1'OABo.
z=f ; .sflsln
az=tf a0
.r,r$ JlFl ;tlll, e,#.rrr 4, rylr J* .",n=*Cb;t .Y Jt'l'
f*t-.f t+i;11 c,-rl d.;ll "'qL;
I *= !.'"lzl-t
JrLi. Cl.r;l a,5Jt*:t . ( lrr )
ott rtruoaeT=lnu
.r,_r$ JlFl ;tdl, c.{?.rr ri. .ylr Jrb.r, n= * Cb;t .f JU,
I * = [,'" # =i [.'* ,-, do=.
lzl=r
r.5c+ti bljA # t* +
Ilzl=r
.(+5.-rtt)
;Utl. c^{-? .,r 1 { .ytr Jj' .tt gQ)= + i@=i et1jt .5- t jl.f Jtb
t'3 :'r''"d Jlt'ol
I z a, = t e'P i{ do=tniItl-r .
,'o iflag I'I zr# = t ic,p d0=.
(z'2.)n
o ,'Z*\
lni ,'z= I
rdzI-
i't-7
.LLirJ.i 6st Jlr(sl oAr.il{r ,i.,t.Y.f
p*t-.f .t<iTfr$ JlfslA bB;l tBVA 2lc ;.,:,,tSsf=f(z) CtUt.6t.,-,ill
( ttv ).... ... pt{'",F
BAI fe)dz=-! f@eAB
I -.t<ii .r:-iL c-,ti rrp gf sa $V . -t[a f(z) +fl s(z)ldz= a I IGlat+P I slz)tldz
I I frryrl sML f-rb 't*T lf@ I sM tc "''';- ''ry J* L'6'^ - c
| ! ypyzl sb b il'r '"
! I fAl I z' (t)ldt sM I :6'1+T dt = ML
e.lj {l r:-,r .g;bltfr f el-y il Lt r-,, U ifl LLi'- Cly ;t .5r3ltJsl ,ry rS-u:.-r ,f.li lJt Li .J::-r ./J'.tc Fltlr r!! $-).c +lt" f J.d-u.."r
\c t dru.-r ,-r4l;; f:- c.,t.r'; .:1 .,/J*c.JJl) \c ,;^itr.Lt c--tg=1 b-.lc
.r;.,* dlJ-t tUit .r.rb. Cirl.u;t c i.r1;r
r.tf J*;J..1 ..rr,-i 61p., .l ,,4 ,.1-l lr b Ctt otl jl ,S;Jl-*l -<-S-,i-tr-..,L, *, loL;., ,jsl !,3;Jl-Sal .(.5t ;i ":,^'tr
.gprrr ,ly slt-t'' 1-tlt a-l t,a*- :.Jt*tJt+Jl-Gl
g.,l lyyut-{'# #.{ t-,rJJlFl # + 6 tiri .ibl ! b Lri t.V-tiot,-/ dtJ^t Cly :t e .$|*l 3-S- q q,.ai orl q +i U.5 ,-,Sa ;Vi
{ f?)at= "
(.,5JlFl d:iat)\4..^at
i 6lil r o63i r-:b ugf (z) -l ,lJs D ,tV J:-.ir e 2l.sslf [email protected]'D ss ilr" u' sa
..:,--l ;tl.Ilr c.1.7;)C r,;-, 'p t rS-#f-3;,il .lT;t +f
0 f@ar= 0 t,*-rayl+i f @dy+vdx):gr-rb ..)L!il
/t-- - \a
s>)t, Jlie UJ #;l o[r lt1
!feta,: Io[ r-X #)dxdy+i{,1 ,# #,*,aV-af@ gb..n; rrtJ-t q.+1; t{
! fl4a,= 'ort;rtr
)'; rf rlr oLs 3l .g-,1 r-r^! dll Vt-S ).-'; ;t;9; *; t{ qJ C,el crtjl
.[-- .pr ,r,-l,rry.1t.5 cgblr l={ Cr_l .r# .ij- Jl1C *5 2l lzf Q) '*Xf ofia,c d1.;r sc tJ(z).6r^'y, ..r.-l,r$-r,i r rlJ..r Cly + Jtj^;,c,l)t(l
daZ ,S'-2. 4Lii rfc !l.rr r, ib f r:--, p -p 6ljl. .tl3i JJt{ ,rtJ-t z. daii
1,,*llt r:U ,.rT ,llt
f n'p,r
\\II
I/
0u 0v du 0v
-= -=-ay ax' ax ay
f f@a"=
4iL h Jlt a;u r;a.-.,-J q .c..,1 jUL. 6|4+.rr I I c J d-J{Jst ,-ri 1'r rf
f(z) ;ny.rrlJ-t.:Je t t €C,!-.r+Dt 3D1 +'-[ J) q bI.l c it'''rr Cr: ilrt+ r .r-,,1 j4 Vil ,?-l 6s4f@ eV JlFl .12)U, V .V J !-ti L?-l J.
C*t-rf .jL;, c.1+ f Tt sf t cg1t.t,S-At-3;,lt
I plaz+ [ 1p1az+ [ 1P1az+ [ fe)a'=.G1 rt C1 rr
f f@ar+ { f@a"=. LJ
cF
t4-+rt c.22,o;t*, .i>ti €{+ tr FJ J jtii, c.+? ic J. r;;Jlt3il..li ;r rfr,gi;r i
ff@ar= -0 fela'= $ 1p1azcff
Jyt;r Jlr*;l g.r1r.2r i ;EL, c.{? tt -ilr Jrl JlFl I 6} JtJst .li;r ef
t-t-d lt !l ;UL, c.{-
z1s...zsz.IoLii;r rt.f rl9.p.r tc J.Sf(z) rrlJ-tCb.sbr Jli,-lltUfy al
f-rlr .t*i -t.lF up;2, tz. tLii9);) Vf(z) o6.r^'1:, .tlt f..^, tl! dlJ^t
lrLi. eti :t e$t.Sr;t
ff@at= glzyz+ f fela,C g. Cl
.tJ-4r./ cty,o;t*, \a{? J)cr Jc. tc t S.JjtJsf .li;s 6
grl 6g9,=--, ;lB l;A tlf@ JtFtJrri. 6.:^iJrgl alr ,g \ r=-ai q {+-f tl
Sc, u:.*t J) tl bB t,4 &i),r ctfp Cl.l ,, .c--,1 ,y;*, JSo. F,.A t l, +L.i;1.r
l:4. Gr.t.s., I=c, +cy ot<ji pS-l e q -r:;(; car b $l-" 5c .lJtt.;: 6ilj c,,
c*t-.r us Jlf.l {:iai I qi ! e .:*-l
t+i;l.r
f fe)at= "n
I qplat+ I fe)at=.ct cy
i/_fJt+ t :-*,t +"V-A bB ;l c, J.,B UA jlc, J ,S-SJl-&rl .li.2r rf
C1 C1
or.rL.ir .t<ji f.^) -# B VA :t S-SJil-3;l *A bB ;l ltcr J G;Jl.frl o6r.o 3
,,frrn 1,,-ljl.r stlrtf^ q C, tC,6t45:. ,t Y :t e.A|*t S €'*.rt ,*: dg,;lr l: il # t.^f ,tllr 1 grr"r,
"r!-L; f ffOat oy,o t b JlFl
...r-.,1 &Cli;r (;;1) Ai q.ai) drt-. JlFl d?ai.:i-.,
,llF.: tLirz. € ,?lr -r-ltr JA-JD e o:L J:-4^ o)r lf@) o6y' .Y 4';i
5 oty.D jl
F(z) = ' flr)*
.t<ii
F' (z)=f(z)
B
! f@a'= F(B)-F(A)A
D ;tB tA &fi f f rSl;l q ,tni"-,r
.F' (z)=f(z) .rT 1: r5
Iz.
JrUi. eli :t Sj.lt.bt ( y. y )
f@=E. zlLz
*r,
F'(z)=limLz+ o
F(z) - f(s)dsIz.
F(z+ Lz)-F(z) 7r/n a+# .rrLz -l l't
d/_lJq G'Q)=f(z) o6y' ,x.Vf(z) Cb d.rt Ctt &- trG(z)
:tryD 9 glsz jlN,-&L*-.r 6_.c,-l D )*lt &d ,fl2 r(rT c-te q .gLJlp*[.f orlr
"pll; z tLfr;o ! ;:-:., .i.lk
"5,-t--" 4l ssz*& VBol U .c..-,1
F(z+Lz)-F(z) =+ [['** f(s)ds-tialart=r I'** f(s)dsLz t-y z. z. -t Lz n,
, F(z+Lz)-F(z) cr-,t- \ ^ z*Lz r,-1,,. , o'ntli2'
| #'f@ls #, I,'-* tfrt''f(z)JdslsdT ex lAzl =6
c.ir; ttl3,-l C-h:tz*Lz
t fls- f(z)ds
eJd d,.i? Jb
ry.fe)= * I:.* ff(s).f(z)tds
d Ji, cs)-t'e c-.l, r-r.e Lgljl q Cr_htf .r.u );.;-,,r-,,.rr-g "[J^;
,,>le u.f(z) gl;
bd J,5l ,,,r. lf(z+Lz)-f@l<e otJ3i l&lcd o61 ,5,&* r., r:-,,t tfr-f
f-.rlr '>*,lztLz Vz ,l 6;J|Srl .lla .flC, jlN.-&1"."*^ g,t-t )l -*-flf(s)-f(z) | <e , lr-r lsl&lcd
G'(z)-F'(z)-- "
D ,y_l;lF'(z)=f(z) r(!-l u. ]t .'>-,,\f(z) dJl Ctr .5,-
( Y.t ).... ..... . Ph.ts
.c-,,1 c4tl 6r-ra .J.ll Cb J) J-;t it ,-p-G(z)-F(z)=c 4\tV s
ttliCD f 2..Lft &- .1Uol !.63i r-l,q D f il, *Ut 5>B tA ;l JV
'.lit2
Ifew= Izfrzlaz+ I'trrl* = ltrela'- Ilel* = F(B)-F(A)=FQ)lBAAz.z.z.A
l'*' = (t+;)tto r
-r*1t..:*,,1 & Cli :t,S,.{Jt.ful .rsLr,rIJ'^t Cly :l g.Sll"*l oalrrt+
J,,L t p -# ,-rst t,fi 1;.F Css !,s;Jl-3J;,l .l9a-.r 6$ll*t UttVtI
.$JrtS !, i !*,-1r, bliC b r'lY
,l-L; d c.-,,1 J/-{ dt(, I ,?s t{:' D -,rr u- c a }=* * eli :t s-AL*,t
1l crl,i:.,,| l{ t{li t+i;l e .Elt-S:rl.l -r-itr \:-,') i P g;Vltb,
.-lfr ^+i Oefrrf+ .llo-t-! .jbl ,-[..oi ../-l J9l i .d tt-f ,J-l;r ,l-tlk' .r:'-"1;i', .ltil
.r*il,i ,/.bc djl tr t c .rr ilJ lrta &1.;r r5r:-"1 ,f.li :t tSr$l.*l 'sL..r'.
.ri;r.n )t c;fJlFl Jlicc G;{.t\rqL; c;L-l ! Cl" orl q$;t +-S
ob.rr L. .g,J.i JlFl c j>u cgrltu,rr-L-, .l.l* c it^t-' J eli cr-l jl a5..:*"1
dt_l.;i jl Ut.ai gr-l .g-.rlr ,f--, dt-tii tSr d,- dbl q jtli ilrrl ffi { .r{+.r
.stJC $lrl yt Cf o +f r::.-.r JaLi'., Cli.l' :Jtt .1L..:
a
)tr
r*i ,
f'" z'dz-- f z"JTo'
b
I f@a'= I feq)dz(t)
(Y.f)
(./5JtFt J-trA.f 4..ai6,1$l;z sD ,gl3 r:a
'p rS1c I u#;O orL.u.-.r r.7b lf@ +rS rft
f par= wif(z)"
s-Z
,.1,lr .t<ii $U c ..r1,r :r:
t- f(z)=*t [!!).a'
>-t*r,l Jy,e;UL, c^1; ic J.6;Jl-t;l l1i.;r 6
erd.sLJl
=rltif(4 +f"W*
.r.ti df=f.:.rr rJL's )jh, lel,Slt.c.-,| F Al'JlFl Uii 6e+r.f JUi Jt-
-h ,,f.r", =# c..L ty ril*e csljl +, .c.-,,1 $.4Scc .*1, )Je y' ;l .li ;lt6 qt r., r:,-,1 t*:r6 ,tr ,p tJcz. .pf(z) 6brlr;* ru-,,J-,li?j L.l f$rl
z f.r' r'r ovl.r lrslr" .lf@-f@ I <", .riJrap :tlic lr-r I <6 +t zrtcsr;rq
1*t-.r.t*i ry' qillc FlrJ:5,.:5rr. 7t tg.t*d g.rt--, t-t<"Jt./b trr
$ffi"t=l {ffUs }rrou =ttte,=s
# 9*=9c uo c
f-,rl: ,ialJt:1 c-,,1 :ltts Ja dljl 1 6.lt -rti .r-l o-n s
5 f(s)-f(z) ds = o"" s'z
f@=+i 0"9*ea.{.2.: j
r-:^ly tsl5iu.z=2. ,-rat-p +tli .51.1,11
f(2.)=* {"P^*
+;tliC iy tsst I Ctt c;t* tr tJ glt.rl& S $C q:.t # u-l 6s: il
F ,t tsst ,$otf ^' JI)rt*Cli :l eV4:-");l d,+"'^ ,'ri s'r')
Jl.p Cc utat C*ltl1;L;;l o:tc-l ,.lr{ *:.-,1 ,Juht.li .l9.2t t ilt f 6c ta
.€,;.Jbl b.,,; JU, rjh, U-l el-r,..:iJJlFl
.r-ri JlFl ri-0./lr ett tf4)= #Cb;t .f Jtb.C*t-.r ,3JJFol J-rt q'?itt
z'+l
I #az=! +dz,nig l,-.=-rnilzl =r lzl =r
dr.,r d;51t'.l c-,,1 ,/-tJ s^,tt.rlJ-t Ctt.J(l- JlFlg,rr5o.r.rLl.+t A
cr-f .C=Scrtil , JJ--r Cly c.,tL:, )-rJ, t) b 6JL' ffi n;li,-t 6gr-'t
'€c.rLrY*l #'ibl qb g*r
,r-l) .6iii J-!U jt -y t4ioW1p 06y' .f 4jril'r
f(s) *(s-z)'*f,f (z)= (\ )
.slJ",C c.,25,o;Ul:r c{? lc J*;r A;rJJtJst .li;:r5r ^. y ;tliCj r!,1 tf -rf Q) & ,Lii lfe) ..lry ,Lts ! +i tt ..)LJf
f (z)= tim fe+*)'f(z)Lz+o Lz
lrLi. Cfi;l aj.tt.*r;t ... (Y.t )
,l5cJl-.1r 1(Y)f)e)= #, {
f (z)=* (s)ds'" Is-(z+ Lz)J 6-z)
.g-itf(z+M):f(z) Clj 6,ta u.S JlFt J-rt ,f J t6q q Jt-
f(z)=# 9"1#: {H* ** tinazf ,.,,0r{''rni ';';: (s-zf [(s-z) - Lz]
'(s)
f-rlr .t(tT r*Srt*l o it.;l rt.ib -f vz;l &u d/-,ptrt' d ;ll.,lae I Jr .lu
lLzO f@ds t< MLlLzl' ', (s-z),[(s-z)-Lz] | - @-lLzl)d,
t;'a o--,' uLz Jt.lu .c,.--,rc ,_p ,-rt J*L tlf@ I .s)! Jlf s.-Mui;r dq qi U.l .tS.f b ;* g-r-,, q .iy .S.lLb.2r c*,! g,r.,, cr.lk .t<ji JSJ,.,
.-L-t,l.iljl q ( \ ) 61Ls .li
.-b..r $) J:'t t.,slii u. Jl* ,* u.(\ ) J-rf rlr;;tf e1 t1
j-t -f cSl;l q t o)r. ."-t3*tD o:L J:*.r r.7b .nf ,6-* (llf 4:-At) 0 4.Ai..r,,1 .rlJ^r D rlr;:/.63i fd@az= o D I, il.rc 4:.-J
J.!, tt
r*lr,.r
( Y.v ).... ... ph ,F
rrr;,-$r; ott' lF(z)=1'11r1a" grr;-1, 6f-r.5!k Y #,r .'.)LJll.
f .: ;, r1L, c.,,,1./J-t e.UF dr-l u. )';Jb .-:-*lF'(z)=f(z) it s try fQ)
.btc/crtil u.bf # g* Or* rl -t.ll.;i ,,rlJ''t c'"'lF(z) )t 6 ..lT ,:t':'
ritt c.rtj.rr, ,JJ-t ,st+u..rrF o6y' .(f*;St, J-l "-#)7 4*3
.r6Lr.tlt;l q7b )y 6Jt l.1,r f;5t JU.a, lf@ I 'tt"i
t' j)';'2. 4L.a 4,.1r 1 J53;L:; | 1r,, ,S.tlt tf,..,-t'f' 2l't';'f(z) o6y' 'gtJl
€ltD,: d €C.rLr;l St* l, lz-z.l <r. 6loy-l.r +=-b .t<ji St*;lD .!-lr
iri;l 6$t-3ct\: lf@lslfl) I g*rl, 6loyb reb itz -P sl;l q '6;i r;j
't'*trr
lf@ld0s lf?) la|--tnlf(2.) |
\rt
I\ln lf@ ld03 lfq)l
U-;
(\ )
f(2.)
-\ ln
_\ flni t
r,
,;SJl-Gl J-rt 2tdz=ireiede ,z'2.=rd0.r)tcr-,if--b ti Jt-
f(z)tuz-2.
1,*trf r€-.,1 lt't.l(.r 6lortLr r'-b )fc, tli.1:6
f.'o f1z.+rdeldof(2.)
.laLi, eti :r s$r.&t;t
I fQ.)ls
tf?)t =+['" I lfe.)l -
\ln
t+i;t.r
!.'o lfe.+rde)ldo=+ [.'o lf(z)lde
1'ot-,-f (\ ) g.lt-rb -l rS.lL-b .r-l t .+-r lr.l
[.'{ | fe) lde
I f@l I do='b-;
6i-Vr+lJlFtitJstCbcr-t rtr I f@l s I fe) | c_.,t o:,;,-rJd!-t *-bC=-lt{ r:-!l: z ,-r .Sljl .-.t ,.:,-,,1 p) Cr_l;Q c.-,,1 .k,ii JfJ.ll ,l-ra ; &-,1
...:,-,,1 +ai €f ;>t; 6-uli ..-+tl ,LAf"
+\ f(z) s*l )l I fel | = I fe)lb g-JU J r5Jt+il rrly,-# c,
"i--l.r .rr l.r :1i ;lri.. r-jt, I f@ I ixfrtr
.S.ll+; I c, cSlt r-r*-\,tC, .rr -trlr* D i,rl;l: &A ,Sr_2. .,Lii g!.z.lsc, J, 6l&ft dri." Jl .tU.+tJ iy .iL:l rr.*r tr r F-r Ltloylr reb #.5 .h )r ,2. rl.rl: ,lj f-,, q ll5lc{U) r36..2L;lD jyugubrrfJt,Jl.r'i, -$lt,Jt{:i I f@l rdl; A
.(r5
c.rtt.rrb f(z) ,€;i -tt Jl.$tt'., fb.*b/o6rn .(rkf-f:l 4-ij) V l.jt'C-"1
$U sJ.J'sc.: I s-z I =t oyl.r fu1r.rt'.+La d,rrrrS.-r.rL:, f qji;r ...ltjl.t*i
7tn)6)=#rq.ffic.-,,1 lyy u-s 651--'ti q r5.r-l.s.rLb. o63i u\lf@) I .s)tr JlfMflJt-
tf@@t= ry€)d
p,*rl:n=\ 6,|.11 u.9
tf @ts +.';"L" Jb .ry;.rt*;l Sttf.j! Jrt dJri lltr Uls6f@ gb rr:y ;b c^!e o
Otr!q #.)ttl :fQ=c ta;l jlsf @) =. ..*tr.f J-SL...J1,,.*,-t+ utt>*-,,+tr
'-L.,s.f
oye r, 61d.7 r:" Jr \lt ;lah:iat) A 4.atp"(z) = a.* a g, + a.g,t +.... + a rt i an*"
S 6* q c*,1 :F-f z. -uiL 6l+LJi; ,J.:r.r-.c../l je,.(,_ A;!lr,Jil)
Pn(Z.)=.
csl.rlr V raf(z)= #.r.rtr.t*T p^@*o rz rrcSl.il q 4?rt?..)til
r, r5..:,.,1 tfrr,R Jtu drJr 6 ,fcrti clly d .tr,.,lyf (z)=- m jr.-
(l Cfu-) .**rlr rlz | >R tz ,,e C\l
trrzit=#.#,*Ep"(z) LJ@ +f.:,-,1 1:lf(z) tlr;a .r$r.i,.:Jr er 9 ,i...,1.r|.t;l.,S lf(z) | qp;.r
It c5t-Lii r:l.pl;l Jry Jb .c..,,1 dld-? .r:7 .5,_ .-,irrr; o.-f .iyt 6r-lt c.:ti
-r:7 t- 4,.rr 6,*jlr* at Alj)-,; * eliJlFl u,,l-., ;"f-q Lr+, aly
JrLi,Gfy jl g.pt.Sr;t (Yt. )
t dr:r,.r tglt lt t it L5t-Lii .r:*+ "rlJ-.t
rli J51.r U g,,$r|-3u,I )y t il, rL.E
q jtJ ,f.l-f crr? JlFl (f)J.jl # c:25,o f .tt.rJF.i r-lJi 6.t\jl.r,rl dl:r?
,rl* ,sV^r t l-/-r ,lV-r tleL;., jt; sV-r optiAl* : t+-r c;Lui
.r"jlrr,r/ Vi ,l*..t ,Ii rr r5g,-rlr ,lbrl.l
ObJ J J3lJ C\-r,..tlt,.sQ.r.f .fJy.e q b.Ji 6zr*2,,*... cr;!..t*i J-it laLi,., rl.tel;l 6lrJbr {zn}r=, .6y'
ifr..t-,,1lyyleLi", rl-tpl il sr.-i-..t &*)i6.; ,: ,
,n Gtf
rl-rrl il 6r- J-- "i,
t ry) r !lci,-.rl-r.cl;l 6lrJb: t ry ) n-,
Ctt.lr.rbb 6!l.uiLr J.Lir 6tj CV^f l trjbr 4f;r d-.*1..:.-,,lleb'.,
Clj lt t+-f ; brJLir tt-f f sfV 6tau, f=.l1r,J.t t5:,-l .1r L .c..,,1 ,r,+dj"-r & Cli i t-t+y.lrlt 2l ttljg c.,ti3l ...Dl ,, p*b .Jbl b Jcb,
fJ;-lr* .dj,l5q t, t#T tLr,- Cly;r I ).15orLi:-,1 ,A.Slt t alrl,o ,ullt
rf ,r.,lor.;9i
guV.,t rr l orr5Lr., l, tfJlFl u. )oy.rt d-i, c^{+ .ll-r, L S gir.tt jl+ t{:tr,&:,. u-l Slu f=sJ- V rS.A|*l c,,)[(:. 6fq ,.*btt tgi'(Jt
.f*lb iti ,SUJ-'1r.l.rJ.,t+-r il gV.5lr;.,r c.il!i'
cVr.il &;
c. +c\z * crz' +... = i c*t
C. +C, (z-2.) * C,, ( z-2.)' +... = i Cn ( z-2. )n
(ltl ).... .. fik4J,...i
cSV-rd,.,t ..rul;rf z. .bii Jf tc.,La:;.r lr* Jf t jli SV-r q .rdi 1
Vi ,j*.i .J1i.rr L e Jj;lr;gy cglols c -..-eljl .rbJ s t-E jli CV.r jli.r'tltA,-t'
Jr lglata z e )J^;2. ffLc. ry-lr irllr gf .t;S uaj .(;3J,; q:at) \ 4.*Li
..,,*llr .t<ji .l^!L c. dltr
f@ -- >. cn( z-2. )n
.li;.1 ef
f(z) tu( z'2.)"*t
't':'"'lC' tJJ;lr S> glSZ' f.t +d;loy-lrC1 j
{G1
6lthibz €g,-f I , $Crt Jr.13;qttj !.r*;1l.rc, 5c.24lf .gL,Jl
ill.l) sc, tj|) rlr; .r$r..,- c-b r.,r< rr1t. f;lr .t*i lt - r.l =, -lr, Jl.l).r, itr-,.>)i rlly.r, rll
f@--*{", ryr1r(r-l \+i!.c,-"1;tdl c.{.? ,u,ct J,rs;Jl.&rl sc, jssr.r!.8 ,S--z.li.2-: l:i
lrl<\ crljl
I - \ *r +ti +.... +uk'' + uo
\-z \-u
'ejdffi
lLA. eti :t ej.t1$t "..1vrv)
t_s-z
_ \ _\(s-2.)-(z-2.) s-2. z-2.
\- -s-2.
\*i 5ct 6t.t ta$t-3,*it [email protected] rJ.rLi u_t ,y'-h ?-t'\l * I . r u*{<!-l
#{,
f(z) - f(2.) + f (2.)(zz.) .rye-2.)'+...(k-r)
.me-2.)k,
(.*t'..r
+ R*(z)
.li;r r5
R*(z)= (z'z'-)klnt f
ct
lt-zlt ll s-2. l-lt-2. ll =rr-rdr-l q qj l.t
.'.li.;r d
r=lr-t.l,fr=lr-r.l
I R*(z) | s
c*t-.r \to.o
fG) =2 an
..li;; 5
Jf r, t-J1- Gf ..:,-,,1 lyyz=z. ,L;t Jy t* uSf I J-b Gli tSf
;Vl.-Cb .r.a !JJ,t<, tSf.Jr-i ;r'y, ..-,, I l-y i))S- Grr., crt-os., lr."
.c..-l or.i dll
I t I (o
an=rYt=*|,ffi
d- \*z.++...,'
slnz-z-
cosz-\-
sinlu=z
t'*r/
,'*Y/
7f
Y!
I'l (o
I t I <@
I , I <€
.f,rlr.t<ii.r-!U D;r I f@ I .r!q i1t'.4 M ;l JV
rk lMnr, Mr, / r r&
lr, (r-0rI ,-r'T'| <r. 6, -fdbl q q# ,',#Lfu Q) l=
( z'2. )n
.d/lJr+*< \r-r+.r
coslu=t+ !*!Y/ tl
*= \-z+zt-+...,'\*z
Ln(\+z)=z-!+{-+YT
Ln (#) =y (z ++. t *...,,
+ ... ,- I t I (o
I t I <l
I z I <l
l, I <l
... t
erlt = \ + I ++,**...t
t.+
\_z' \ +y (z+\) + re+ \)' + ... ,.
d+'=e[t+(z-\).++...]
.rfc^ttt JljC rJ.tu:.r
I z I )o
taru = z **r'* *r'* # z,i ... ;
* = ) r\rz+ +. #....)= 4 + l- + +. fr * -.,sinz'= \ _ z' *r'__z' z' yt o/ '"' I t I )c
f e+\ | <l
I ,l (o
lz
I t I ).
i-p gu \-t b tdi elr :l it s f-tfCt .r.-t-k Cf-r q [.r+, ir15l f^
pt{A J.ai
.c-:'i Jli,-l.tl3i J-:q .r$'.t z. 2;fQ) 4?Jt " f+r./
f@) = ni.an
(z-2.)n
.t-t.:,q $s*2. lf G) Cb.Kr" tan =
f ( z.) = f' (2.) - ... = f(m'') t .', ) =n, f^)Q) * i , {n)
1 z.) ,.ri.2r ef
..v|6f @ eV {^ i-;-t P,S''-lrz. ot(!i
gbllm Ff P S.-2. 06y' ,-^,-..r.::.-r Wf (z) ;t;;6b g;bri-, .Y 4-;i
3lz * z. -f,Sbt q S,A-Pq c-.,1 )273,,2.-;t.,rit..-.r.5-.6.ri #Ul @ JI)-J
.f (z) * . ;!.'L--.r d,,,1
f-rl.:2. ^u,,i Jf lf7) Gb.rrl* !-'.1.r ..)tifflm* c.An - or'n=\ r...rft|.'\
€
fQ) =2n=m
t>--,12.;:.$-; *Eg (z) 6 €CoJ-oL!. I Q)= ; an+m G'2.)' ef\
dJi, 6;tr-eec..l, )bf 6,l1tu.z.s>BQ) urt*;:.'F g (2.) = om* o t
tr.l g @ -g(2.) I <eg,-"rlr lz-r. | <d\2.*gl.ii q S,h-pr,.:-,,1 stry
dlil q.t<ii f=fr./ r&L.r d. r, b iri t{ }lp, 19,6 6,*-c 8= } to. | .rkl
1Lj $.tz.fttr,{t*.nJb I S e) - s^l < \/f I o^ I grtt lt-t. l<d.
Jtl ;5 g(z)* o ,r(r-1.'"g. gtl ;lz*2. -f t1lih; j .bl .b ,r-rr$-r.-." + lo^l
JrLi. Cfi;t sAr.*r
,S.lL-b iz tSV r,,.z, otfdildq q.KrTg eJ =. ,(,-L*,^ gl 2lz=4 rsljl t.J-', ) 6..itil q {:ii 4--riJr .c'-l .j)[; 6 tl 6l o^l . + | o^ I ) -i,-t'rr*rs jr;
.g-tl>e,-f ibJ tSr ,S-,- 1.1': +5 .tbl ! ir.1lltl .'r
D ss €lsr.,'Je,c;.;r -r+ Adi :-trD..-b tf e).6,r.(OLJ d::;j)f 4-"i.t
.Kii J-tU,*l*t@
f-r. \ct tct ;Sll-u:. oyl.:1r dr- {,iL;, il.l 6ltLl;z,l c-t p.Kr-c .li;r ef
..:,-"1 D i eltr.
't<ii f..l5.F.r: bcr tc, cy-lr.r) dri {iL s: g\z fy *f .ylr o6y' . .}Ff
f";lrf 19 - f try f f6)tu =.cr s-z ", s-z f s-z
f fl'=)=^ - f rytnif(z) =.cr s'z cr s-z
f(z) f(s) ds - \s-z lni {
cr
p*t-.,r 'F Jl-Sol J-rt q qi tt
t+T;l-r
(\ )=* t,
\_\, \s-z s-2, (s-2.)t
f(s) ds
\(z-2. )-(s-2. )
-J=s-z
= \ _ _ \ - s-2.=* +z-2. E-2. z'2. (z-2. )'
\- -z-2.
+... + ( z-2. )'k rTr+* .JJd!.r
ff)
,s.SJtf'slr! rfc 6u J,s;Jt-3;;t:f (s) tu ,o1r)r (Y) csl+-rv cri-h vtiu.g*t-.f (\) cSrU u.ai tr.r J-b cSts-rU Cri.b C:+r $-tl.tcr tcr.tl
f (z) =c.+cr(z-2.) *cr(z-2.)' +... + cp-r(z'2.)k-'+ R*(z) *c-, (z'2.)''
* c-, (z-2.)-t + ... * c-1, (z-2. ) -k + Q* Q)
.li;r rf
lrli. cfi:r e ,s.lt.*t
Q*(z)= *r*+R*(z)=# {
ct
fu; n=o, t\, tY, ... , t (k-\), +k
(s-2. rk f(s) ds
z-s
f(s) ds(t"z
c,"ti.lly4t3*;f.rrl*+.ai1rd.sLrrr1 ( r 1rr.tJl;ir= l r-t. I
(vtl1
t,=# f,
{C1
t6;r
.rr5
te*@ t =# (+)r R1 Q) {c&*o
Lr-+J
.J-2Lr..)L,il q .rbrl qJi rrr;i.;) I Q* Q) + "k-o
e.[l,a ii .2s tSz. &it Jjztr .lb-/ 6r S--ottn uJ (z) Ctr l- l5g-,lsrc_+
r-rr;,r$r; *,JJe u.lti.c.^-J./-*.Jt<rldt #q qi b c,".,1 Jt;; *fl2),-,.51 :l -* dSl {-l ,.JlJ+ .6r^ Jf .e;.-* ;<., c1 12'.2,o r-.rtr.. z. g f@)
C-, ?te ,-P* a "c2,a;Jl.-J"
*lr;.ll.lj cgf --l* r..rLr.r.5,.$lFl
dtJ-r rr Ctr JtFt r+.,i;t rc-r = $ f" f @ fu e.\,-t.r<iT ,r.i q-,t^, bf, f(z) flz = lni c-, L51Lj 6u )l c-,,1 .lT ,utr $;t ,5.,-2. rfc Ft ss.t tfe)tsloj'-s c -"-rl jlz. JF :tf(z) Crt irbJ Jo-r;r
*n*Lc-, ..1.,1 r-.,t-, .[rti.cillt L;LJ-,J* Cl-r JFol l-,t-., tt tslj4 uT r..,,t-, ! fy; r:,..,1 jr.^f,,Offtt i .J..rr.rr gi-l'.: *: f(z) o:_* q J J.,bd z. ,Joii lf\) 6b rJuL bc-,
f(z)tu = rni Res f(z)
f-rf:f
f
ol rydzlzl =r o
=+(''-+ z'+1" -
cos zlru Kes -a7 =z=. z
" t'rtra F
.JJr...L, lt $..ttJl-Gl;l 6*,r^ .\ Jt!
b) I z' e't" dz
lzl 't
ric @
+ t= I-\t\+\z-*...t "'' z' y z f
rni (-{ r = -:"r+tr rq"i;lr
gj'lgrr Jlr3l sl-rtfu
ry=+cosz
F COSZ ,J 17 Gz=lzl -r z
z'e'h =z' (\*
Res z' e'h = Lz' c f/
! z'e'bdz=lzl =t
+.+ +.* *+
-\?
.-.)=' z' + z+ I +
iar,51)
_: t+i;trlni Res z' e'h = lni x + = 'tt
z-c ? r
JJ +...f/z
if, ^t t r? J) ef c;15, Af,r$'u * eli JlFl r-"t'-' +'tli.l jy st"r
.€C.Jbl b Ji # t.F Ul Sl.t,'r:-* ,JJ*t Lsh:-.l ):'ott> i
rri rrj;.r.,r 1 "JFc r;-r 9 'L :y i Q) '.lSr;.(Lrc$l'r bii ) f 4-At
{" r@dz - 1ni > ::; !ol_,::] :u u}-}';z' "'' zt ' z'; 'r'u;',r ''+ \
7t 6t*ltzn t ... ) zt )zr Jzr fly \cn t ... ! ct Jc7 ic', oyl> n'r)t'jl
Jl:L,g.t<jT J:-lb .'-l,l$,J-*&n rn ! .l .rt c {-**{ (;'*FI, g g {g
I fe)dz = f", f(z)dz + f,,flr)a, *...* f" fe)tu '*-:.,;
-1ni Rez f(z)+ tni Rez f(z)+...+ tni Ra f(z) =tni 2 Rez f(z)z=zt z-zt z-4t z=zk ; k= yy-St
f z'+\ dz
Vl-, z(z+ \)
bf=, #42=1ni t::.#.,I:i ffitetl*.eot-.r z= o tLfr.rrl.r
Cb orl.ll.rrl L*,2=. +LE g#orjL c;ll \.sfir*-1'Y,Jl@)=#
+ *=+ se) =)r*.t *s'(.)z+... 1
JfFf {.:-,,t^, c--,1 t-t3lbr.Y Jtlr
oo:
O*tft r
(YYt
\z+\
Rezz--l
zY+ \
...Ph,F
Rez \
z=2. z
z
\
zt+\ , ,
-
-- Et't =z+lz'+\ | = \z*\ l r-o
=* he) =* thev+h'(-t)(z+\)+..../
z'+\ =h(_\)= z'+\ l=-rz . t z lr__,
l'tJ. 4r=1ni (\-Y) =-rttiz(z+ \)
= + J+iu;,Ja:r.,r e
t+i:l.r
d/-lJt+
'"+'.\'e2h(z)
z*\
0lz l-t
1td. jt* 4 rSltJ .r-t-k * eli Jl,,(:il r-,'1o., ,Sll ;-'f o'trLi"r'?Jt+
,5l5+,9 o;by P ,r-f ./J*t * I'V'.lt Gb url o'riL -/-)U' u"tr' + bly,-l
* elir$L /-)U, r--,tr., rf c^".1 c.€ ga-\ .c""LJc"t JlFl l*L' q Ji
o29,o "fiU
t 16grl r;ly :l it;) .c-'l sl\-*J .J;l'j'"-r c--'rl jl ,/J^t
,F;-{ gr-rlr t+ t.l c-.,1 --tltf @ eV tr Lt u. i'':'dLry.*lr \t'-s,r.f
f+i .rr-b., jl # J.l .',;lt X.f @ Cli :l ,-fV {,:") c9l; r-ruL trE,. r--"le'r
+"i, JKd,-a* cgl; trc.lut ...rL, q blt 6!> lr d f:ll,-l lt ti *rut^i
.J::.-l
a-l if Q) ,6jstf (4 eirri:, {tit .51 t-.i"(; d'n ..('- lrz' tLit '\ 'q;*l.$ti /J-t &it
Jistre u.l2z.O$ &;t Jy,pf (z) Ctt i,bJ L-l'6r^ rY r.4fl
lrli,Cli;t eAt.S-r;t ...(yyv)
@-€
f (z) =
JI4l c--J b irbJ Ju-.1.: '# Lg$ty ! .r>t-; ;^,_ cn (z-2. )n .L*i
.&b.f irbJ L-t
JbJ \ *l i.....i 06y'.r;".ti r.,Ll ;,5r &Z S1z. o5r &.8 .f +ft..t.ib .l-? ,/tr-ti rl-r^i rgblr
b.c-n --l-er-.n
o6rr *.bf(z) gb 3ln q ;,:J.i ,Sr-l.rz.O5-r &.ii .f t{J*t
"r -r-l! -F lltz. tLis Jr,tf @ ev lLJ L-; *1 ...-,i;lz > m 6,ljli
'p-C-sn* o
C-n:o ifl=m*\rffi+Yr... 3C-6i o
.r-lL i: os-re g,z. JF lf\) ett.rbJ L*, F-r .r.r!r q
f(z) = 2. # +... + # *c. *c, (z'2.)* c,(z-2.)'+...
.eb.f o.rL .-li ..a b Jj uir, #.r,
.ll_rr z. ))ltf@ 06y'.r;".b u.;ll.l.rr drst ltit.4 lrz. g.S;.,L8 .0 *|J*t.r1j .J.,-rit. .1r ._$* d^,tt nz. .f ,Ju..r'rf Ctt.5r.5*irrd Sl*
P.--L dr* bW Lt.li q r+:, tr r -r.:k ls t;gl, jl 4, -f ttl;f c9r.f JU,.r;5
ta) f(z)=z' e't'
d) f(z) - th ''z
-o
,l-- I
b)f(z)=# c)f(z)='#
( YYr ).... o.... ... pt{t .F
.c.-,,1 O-G &A 6,t-z-- " i-tt Cl-r r'e'r J)
,' r't"=r'(\+ i. + +. *i + "' ) = z'+z+ +i. +{ + "'
.c-,,1 ,;,!lg=* ,Ld 6'-z= o +ii '*-*f=* cosz'= ir''-lr'.+ /-+--.)='i +r1*"*.-'
.c-,,1 f+, q;,:J.t ,fi z=" J'(; e Cr-t :tf
ry=i @-i+..-)=+'fr+"'..^:,-.,1 orL ,- Li .5r- t- J-ll q-t:J"t Q z= " 'P-'f =+ e i*-.--)=\- i.-
z=. ife)= ry e1f(.)=\ c.,.,y.a !z=. rti; .pf(z) = ryd-r,.ib.rl> af@) ;*l:rt O.* &A .5.,- P &i) Ortft+ '-r-! -r'rlr+ dtJ^t
3
Llo$tr y.sU, e.-,,L-e .f .f
bz. ifeJ o$L Jl.; ,-l FL-. otJl,ii r-rl* lh q-t:J"i,Sl.rlr z-.ttfQ) 06y'
(z-2. )^ f(z)=c-r(z-2. )^'' c-r(z'r.) ^'' * -.- C-m * c.(z-2. )^ + c, (z-2.)^*'+ "'
1+# s>z cgV u.z. ts>fct;dt* tz 1r.c.-j.lqb al ,>i-h lt 6,r9';;-'-.rtrz-\ !
(m-\)t r-,=:,:. # [Q-2.)'n f(z)].f*lr.r JeV
(VYT)
t+i;lrc-r = Ret f (z) =
2=2.
c-r = Re.l f (z) =7nZ.
zt+ \
+Em(m- \ )! z+2.
lim (z-2. ) f(z)z+2.
#[e-2.)^f(z)].t<ii u\f(z).rL +"t 1iu-z=2..6t
Re.tz-ti
Resz-'-ti
z'+l
zr+ I;iFF
= limz+ti
= limz*-ti
(z-ri)#=
(z+ri)#
limz-ti
-99t_l-.-ri ?'
f coszJ-Elzl-t z(z-fi)'
I z l=f i.ralr.!.-l) 2)z=1t lJr *r:.bt ,z=. 'L \.Lt .sblr ffi g,
Res '!tt= =limz cosz - \ 'c"'lz,= c z(z-n)' ;;.- z(z-n)t ,tl
r!,-, ffil2-r", *!, ffi=pdr)+)r =fi,l
f-llr g t>,"|'2=t Yi oL +"t 1r 1g!lt Gb Otl
.4.Vbf@ = # 6b 6Lr..uL .\ Jtb
zt+l -r r
-
=
-
= _3z*ti ti Y -
JlFl r.-,,t-r.:,.,,1 .r. ;lL .Y Jtb
d/-lJttr
,-friiq?) sp(z) Ji.lr s c,;i f@ = {& rszeq i,t', bfe) rf fb;r-
..t*iq '(2. ) *. jse|. )=. tp(2. ) *. 5 rul*l,z. .2r .,$ai
P(z) = p(z) -\P@_q(z) q ' (2. )(z'2, ) * \ llq'(2. )(z-2. )'+... z-2. g(z)
c.-',i ttl3i6g7)=q'(2.) ,So-l q r+j \ 1o-,,12. ;r ,$-.i "*U#
fe = *,r # * ( # )' l,=,. (z-z')+ "'J
i?#.1.r 1 c.rl crL ,'- Li crl.rb fecr-hUr
Res 4 = fim (z-2.) 4 = r{1.),z=2. qQ)
"-r. '- -' / q@ q '(2. )
...,;lr .t<ii .t.!U .rL .-Li cSl.rlr z. ,sf(z).6.t ,-f-
Res f(z)= tim (z-2. ) f(z) = tim LILz-2. z+2. z+2. q '(z)
.+t*z=- \ tz=. lolii :rl.tf?) = ffigb 6g;teo.riL .l Jtl,
rD-- zt+ I ,:--^ zt+ \::: ffi= y *rt4q1 =t
E^- zt+l - ti- 1:+l - \I<eS -----*- = lttnz--t tr'*vE : :y, \tzY+qz,+ l r
,ilt ,-'. t1.r-AlfsJl <J e' ui";>,Jt+llF,Jl ..-,t ., .O.f
-hLi,t2 25.e q ./-tJlFl r...tru . \
i." f (coso,sino) do
dz dzda=W=E b-
coso_eio+e-ie _ \ e+)_)=I1zr+\)YYzlz
sino=ry=+ e.+) =+ G,-t)
f (cosl,sinq d0 = + f' Irl=t
cs,tq .ri r--,,t-., 1.6 >-r,1..1,,-tr JeL;- JlFl 6- q utl '+ JlFl tf-l il
.tt-SC r,.-,1-., '*
JlFlJh,i, Lro$L
JlFl r-.,b., c-,,1 .7,lt, .\ JLi,d0
ffiodeY
-='-
,h-cos| i
f*t-f ,sl,,z=do -P .# JlFl cr-l ..-.,1-r,Sl.rr
uoag:f,i
d/-lJtr{
//+ e* +), + e +)t +trtI
rdzr-
l/l=, z'-l'/l z* \
lra
lrt
(YYv).... ..ptriJ..i
+(rollr Fl, ,tz=fi-\ +.i tf.r r5g.',lz=lli! \ ot-,:J"3 jr 4;!l:.ltJst"U
,'o d0a
-=,
u o 'Il'coso
d/-llt+ .r:-,,1
=-+ (tn)=r,t
'il ffi Jrts"r .YJU'
Ja'
-tni'+ ft_,..t5..-t-, l.l t
f'o de - dn o y-lcos0+sin0 lil =, ,-tr* * )
= lnixl=tYt
fltl-r
t =lni lim,-!6-i1
e-+fi-i)) ( Y
$4t)t+?"-VUl- Ycos0+ sin0
-,1ni lim,-!6-i1
0
I fg)sinu dx , I
(\-Yrz*?i
c9V te jl 6- \'r i,Jt+JlFl ..-.,L, .Y
f(x)cosu dx , !' f$) dr
,Jily ePa>.,) ,.:,.,1 .gld.; g jr '.:-j :l SF ett ,5{ t>y,e +f(x) .1i.2: e5
r.-,rb., aSl.rf .g*+ ,* 13J Otrt: g-1* i r:,.,,1 ,.>25,o a>.;);l ,;-1.: J>19 i.:
JlFl r..,1r.. q + -l jK I f$)cosu dx yt ;; .rt+JlFl;l &rrn
*I o-L)Yrz
"[il.r ;rz, = * ft -t) Qs r5.:-., lz,, = 1 -i Jz r=] f f -rf .rL Ji 1.r 4;1.21: ilFl Ctt
d/-l/l+ .r:-,,1 r.i- o./-l:
d0
ldz(t-ti)z' + ?iz-\-ti
lrEi, Cfi jt a$|Srt
t,r#JlFl J.tJ.Jtfst J,-t ii+-.Li.i,jlrJrt [-- f(x)eiu 4'
J;,ve=c gUl q &l JlFll r:,-,'l st--rz''JlFl il;tl ,Jtt-f c^.. i
.t-*,-l
-rr -,# or, ,flr), tu Cu ;l I- f@' kn 4' JrJ=lr r-"La'r csl-e'
..,+)F,J,.r"*'-tr+ ':q ! c;b d Lt"' ,-# ' $,lJlFl
f(z)e i'u 7t f(z)e
* dz +
(P-rlr
f(z) eku 4"
6.\r5tz--x , cr )lJ r(+t *. P
f(z) e iu 4,f(z)e'* dz f(x)e * dx +
.rrB r.rr .:-,1 ottaa>.1);l .;.:* l-l.l r, .JpN fl4 eV eP {>.;) 1(l-l u. p
.6,A* q c^it{ Jli€,M P
I f@ I = # 'c"-rv tii,lr'*f -pr
(YY^)
ICy
tC1
=tC1
R=!.R
I
I
Lf f@ekzzdz I s # xnR= # ^]j
orht:
I f6)etur& = nn t f&)etudx= lim f f&)etundx
-€ R.oo -R R+o c
= lni;87 +alo ni tt
Jrll jl
O
I f6)e tu dx= rnijvjt .,L. d Jt
-f- flrlros& na- Re {r"j,ri.i:, f(z) e tul
-f- flt)r*u dx= Im {tni
ut,t '^^' d r' .r.Sc+t, Jli rl r.J*..
!- f$) 4Y= 1ni-o ;bit cj-t t' t
{ #:dt :JrJsr u, n.:,...,1 -."b.f Jtb
i#dr=+ { #nv-*',',:,._-?:* #f*rlr
:r:-,1 Jift3r a;4 FJ) Zr=€ Z r=e Vlf 1r g;!lr \ *zt= " rJ.rL,
d/-lJt{
.tJf
.lat
F6t
\*zr I
T I z=zr
.w#=;+| ,=r,*
\+i
O!
oI-@
limz*i
Resz=i
limz-i
eo(iz-y) =_ i(z+il' le
ic k(z+i)'-l(z+i)e b _
(z+il'
@iz
f e- rlz=ttri x -l - 't:- (z'+\)' "" '- le e
of cosx , ltr
-ra
= -:- (:'+ \)' e
4+J JJ
d/-lJtti
,pV.SkJl.,-(sl ,it rr,lqr .f
'+ #t* t 'fv.rtJlFl ;l it r.-,L, q JU rr" .jbl U c.--i Jrl t:.rJ-+d oye l"Li,., Cli :t e$l-S;il .S,Jq tJlFl g,,-l r--,t' .r.'_tbi,-/
( Yrl
[,- ffi * :Jrfir r-'''l'-' r:-"1 v;b 'O Jt'l'
tiot*4rc a;.-, F ss: iQ) = # Cb;l JlFl crrl ..-,L,,sl.rr
.C+r.r ,b P q br.r g"-lt+.{ bR ,# t $e Jlts"l
f (z)tu =R
f (z)dz+ I f@dz+ t f@dz+ ! f@tufCfCp
-r!-R
Ic
lni Res. f(z)z=,
=trir *#,1,=i tni(gt[ )=-+ . Jt'+t- f
oJi Jli Ccx f tj oy-b g; 6u -t . s9sn r(!-l t .?i!
I Lra I =,ILTNF srLn R
r.rdl -l)
r@e ls t#n^I-:"q:ii 2t 5 | Lru I < lLn * ,r', c, fsiSoyb g* 6il J.)j"+&
f4)elst#o, I ;:
ITCg
ttcf
,s-h tl
JrLi, Cfi;l s$t.$t .. . . ( yry )
i re)tu = [-'f (x)fly- f *
.L\,rdx+in f^ ,&,=-R _R u, (t+rtp r _@
{+fi J)
[, r(z)dz=t r: #dx+in i #=-+*r+f*rlr t+ri;lr
Y {- #dx+h i#=-+*,+
cr4' = ^rbilJrr *-t+ 4 u q l.rR t,Af J,(sl y) .* A:t tf\) =
--
lru r\+( (-: r
(Yrr ).... ..ptraJ"ai
{c
f(z)dz-
lni Resz=in
.t<ji c=ct*cr*cr*cr r $! P tt-r Fc c6.r; f=lr.fI f@dz+ I f@dz+ ! f@dz+ I f@tucr c1 cf cr
4 = -Ynicbi\+x
f(z)f,2-
lf@ | =l
lf@ | =l
Itf@tutcr
Itf@tutc1
RI-R
Icl
-R
I f @na- !c-R
tim tR-ro C
J:,k,
-dx
\* e^
,l(x+wiy-TTZffi nss- -'u"t'li {: # dx
.{d*rlr qi lcr tcr cg6.* tSSs J e .h tl
,I1n+iy1 , , ,M ,(L-t)R\+;f,rfrs;rlf=T;:r.Ipn+iy1 o -/.f,.
T;aE. r< W
=trt(#l:_.o (l't)R
S Y.Erfui+ o' \-e R*o4+.;i J)
f (z)tu =(\-e'1rtc i1
!_- # tu - -rn ietn i
d/-lJt toI
-o
el' )- _ -rni ,ro i _EaE=Giffi= sin), n i. (l <\
JtFt .V Jtb
€,-t o.l, $cJlt'ol J) ;* tsu ti|) = fCr;.it.t*i c=cr*cr*cr*c, J o)t -h tr-r J#c
f (z)dz+ I I@dz+ ! f@dz+ I f@tu
o6rrr*o JR'+ €
f f @fl,-c
slnx ,
-dxx
I^ +dr+ Ir*cf=f'tdx+ I 'i'-Rxcrz
Ict cfcr
dz+
cr
oit
, dz=. ,(\')
.srSJl-G I .F ! .-t ,rJ t. tsliiu, t Urrrru,^t c.{+ r., cy cy-hr.i cgtt JciJrr;.p 3 r;rr
ry--tl i*t,,{*= #*} 1,,5*
tlC qbs t+i;l , le b
1 = \ f-rlr c; cy-h.---i rgtt J| #t *t+ !,, 1a,t s(*+Sr i:.
Ita'=ltarz ' c,
I I + dz I s?Lr.y
f-rlr c1 cy-lr 6 G-t.t.r .y! tR-o Cr
4+:iiJ)
t.tl jl
tr +. f=5 ,-if (\ ) rr Jt lrst 1',
= ,ti l+"1 jlr
d"._(1.-.z
f {-t dzJ.,
clLfl7- I $* I l"-t /7--ni+- '', z cl z
s;*,az=o ;)jqru +Gb.l.:..,| .t*ltJ&'* Liil +cb.tllz='ric^""| t:+-rM Jb .St'r' ,.;,9-c",l.lllf eap ;13
lr,+
t?r
l.fcl
{:+
t")- at 1 s (M x *r) ;.
C*tr.r 't<jiR * o
sM
&-ni=.
4.rr.ji r) rJJ
o ^txledxu-- x
O
t
.- sin xJ,o
cos x' dx
f- cosx fu= ou-* x
dx.rt+JtFl A Jt!
dx=LY
Jrr.ttJtf'rt q rfb
Jr.Ei, eti :t S#.1L*t .... ( vrr )
45r*,1.-., ,l;l1ff
fc
(r )
limR*o
, ,liQf--
YI
cY
,'11fr 5 z=r,li ,, p cgst J
G
.tfrbI
crIcr
++Yl
e-R' e-iR'
-ri,liR YiR l< ,#.*, .-) 6
R*o
+ r=r ,R' lcos$+ish$)
z'
,*,-t#d g--, 1 b R (# t f;'t' JlFl Jr -d ,su tf@)=eb' Cb.;t.t<ji c=cr*cr*c'- .6y' f"nrrf
f(z)dz- ! f@dz+ tct cr
,o'd, - r/i f* ,-"
.Yeo fl2 --
,Yd(eo )
Yiz
f@dz+ I f@dz=ocf
C*t-f t g.#,-l JlFl tr,q tt,F ,-)tt J.crJ.
,li R
RI +dzcr'
c-h :t t
( vrv ).... ... pt{e .J..ai
-rr.rl + | =+d/llq te'R'si,vo< \e sinll2 o, . < Y0 3 L13-l !-b4+t'j
| ! {'arl =} 4=4 ..rc'-r, z' t- Rt f fR R*o
d/-ltt+
tim I *'dr=.R+o Cr
,*b-._r otJiiTR** € ,?t (\ ).rr Jt ;u
[- , b' dx + ,#" !, , b' dz -,li f* ,'" dr= o
f-.rlr ii eL, t .+;, tr.r
{ ,b' dx:,li frn dr=li+=#*=+ q++ qJ:
{- , b' ia - [.7o, xt dx + i !.- ,in x' dx
oeht+
f cos x' fla : [* sin x' ls -- + tt+YY
cJ-i J- JiL- .7.f
JlFl r--rt.., r:.-,1 ,-rr$, .t
{+:i;r .r.2lr I z | = \ cr,,-l: Flt.z)Z=-r- pt F:.:5 .51 kJ .ltf:lt aU
Iltl=t
dz= icie de ,
i9G,=e ' ,
l-.rt.rr r:,.rrl *r.rrlla, .f
.-t15r.-rla
,=ri0
d0=le = y(i- l)
cotz ,clz
z
Ilzl:t
Ilzl:t
cot zz
I, #= I.' ''t1
g,-lJq .c,-,,12=o ;) fJr €l:J"t gl.llr Y g,
j2=1ni Res ry =tlvi lim (2, # )' =lni tim (cos z)' =.z=c z z+c zslnz'
.+5..-,L, b .[ -L dz Jrr*lr .rJzl=r e'-\
*=rniRes.* --riti #1,_.--rtti*, ,_. - rtti
;f:!, c.+? i s jbl. .^i4F J) CIJ jUL, crr-h-; g2.t -t lt ! # .Y
i"!e do
.e'T l",0tt t-
= i I e'
{ - # dz JlfsJl q"'h' r:'"1 --'rl!' 'o
-lni;B;l+&t't
, drt
e'
r(P dxt-t-- \ *'rt
.rJ :71l: l-t-?Z=ere,
\w-;)
--ltti
IuI _ Yfiar\-at
,' at (\
Nll ' Ult: -Yorr'
tLe'=i + \e-:i +l r'T', ?- ? ?
t.JJq-,b, b
)-+cos\x ,N*
OJtJ?eI fu dx- tni Res'__ (x'*l) z=i
(#1'=rttilimz+ti
=ltti
rS-t .lr rJtJ .V
corY e d0
\-Yc cos 0 +al
=+ e'+)l,cose=+ p+)), ot=f
Jrfi.r
.sst>x,-r t: ffi* r:-,L,q
!" = =ttti tim [(z-ti)' !" '= I'l) t?lr t'v't& , rf z+tt Q'+Y)'t
,:- lid"(zw= M=lunz+ri 1zlfl' \?et
cosl 0
gsltz=e' t &t \
fiLi. Cfi;t SA\9t ..... .. ( yf , )
r'o cosle d0 \ nr ' -:
Iuo \-la cos|+at yi frl=,
(z'+ t)dz -\z'(z-az'-a*atz) yi
ir-lJtr
f - (z'+ \) 4,lll =, z' (z-a) ( \-az)
*
ti-l )l.J;-.l r(, 0.1,,-b Fl, ))Z=c 3Z=o,gtC.L: ta;;
!
IT
r
I
@,r!'. -€
'W=\-la cos0+al
lniz= ara
tlim
lnal
zt+ \m,+tim + ( tl,*\
= )t=z+o dz ' z-az'-a+o'L " \ -a'
+5-t=,lr,.lL:i
O trf x'(a|
--
:- (x'+ 1;(x'+f)
x'dx(x'+ t)(x'+r)
lniz=i , ti (z'+ l)(z'+t)
ltr
r--rt-., &.1 .-rrlt .1
z' =rni(4-Ll,? r,
= do ,ir\
Ilzl=r
dz(z'2.)"
t+i;f ,fu = ie iodT f*t-r z - z
Tlzl=r (z'2.)n
dz lrt
-i I. e (,'n)io d0 = [ " ; nt\
Ltni ,' ,= \
.)b-J-t.V.f.li .1, 6 t, f@fu r--,tru r:-rl .7/t .1
z=\+i Vz=. ,ly=yt ,-r+c, f(z)=Rez (a
J-f.r o1lsc, f(z) = Itl' @
\+i tt \-i;l c*l.r L;65rf|) = cos z (c
J.;..r! &J*.;: y+fi U je j f(z) = Im(2,) (d
I L f@dzl s ML.t<ji _*\L J-*u.c ;t->,zt:lf@l s U o6re.y
.t<jTr:,-,,1 J9l g-r.rrr5t)Lz=liVz=y;l lz | =yi.r,,-l: )lrr1,c.5or;.f
r5-r..rr .lU: -r.,J JlFl r5;i.134
dz,I
zt+\ | sfi rttc
tfc
LnzTfr l<tn
.5 L^, .iL':; .t<ji !!U lz I =R> \ or,,lrc o6rr .f
n*Ln RR
.Jsrr. ,b ,".s-.-,, u. $ t# dz ,lEJiR*o Jt,
b-a r2lr1 JfJs5f-U, .5 -r+, .iL:r,r^AJ-glr .SfJfFf -J
! a, C;i U. .?
* sVt-*f ;l 6- -f csl-t"J! Jlt'.t<rT $q f+fi b . jl F oru't .6y' .v
.+t=t
Ln(z+ \) tu
.r:*rrl
4[c
b)lc
dz
z*\
= f f(z)tut' (z'z' )'
q.-t.!q u.V *srl]*n Jtlt f 1t>2-tR rl.ulrS.r c-t rt-b lf #,f; .l'
lf@l rf+t-Ur -r,r$ -h:rbt) 6V flz)*o, R.rr t+ {-rrr) +(!-l ,f.flf@ | >N f*rlrz .,l;1, &;t g g;lil q e c*,,|N f.-:y rl$, ,SLlrR lr .,,3t?
A A.r: d lf@l3A ltl f-^.,,Ir .*,bz J^ dlil t r5-r-llr fb Ctr/J$"ri .\ \
tz f gl;l q \:f(z)=o r z rr 6ljl q t- 6J.r.r .lLS .c*.,1 c..1, f c+ti d)lt
Jt .Sr=^.r.lLs .rt
p(z) = a.*a g*a,g,t + ... * a,t I ali c
r, c.-.,1 sFyR Jtu '*:& otliiinz \ .li.,, +f$q n +tt;l 6ldq -u." 4
f*rb It I >R \z -* csl.il q A e*lp@l>
J-iLi c tgt:2..1 .r-i! .rLt^rc r;-, crL 'P
tSll l.Jl.lr tf -51^5.f+, iJLS .q
.t(ti
ar*o lf@=avz
.li;r rff f@fu u,,tr.r ,.:,-rl .-rlL, .1 Y
.ra.,l-lir Yi1 I r-\ .r-fl !.f;rl )-tc i@= T tU
.c-,1 +irtl.r-fl \,*ti.tc i@= # n
.,.a,.,1 li rr-fi r t-*t,.rr., tr.jb )t c .ri.2.r 6 f(z)= W O
.c..,,1 lz-f l=Y ryl.r c ;-ti 2t 6f(z)= T ,t
(n+t) (z+\)n ; lz+\l<\
(-\)"(n+\)(+r ; v-Yl<Y
J=z'
O
\+,tr t
\==\+tjz' f Yr-,
.5-t+, i,Ls .\f(a
(b
..r$.-ltl .t*i . <lt | <r .61 @
\_lz-zr
o Jl-rs.tt FTin-a
.hrA, etijlre$t.*t .. (Yfr)
.!-.r-l'i c,r,) a., oJ-i orlr tt8 Jf ttlt t;,gl-r jl 4, -f t* c9.;,, .\f
.r.r!t n glijl ..!, g JIS- 6r;l {l .l-? r-,, .\0
c) f(z) = z sin z' b) f(z) = z cot z a) f(z) - / sin z
Sl I* ibJ tSf .;,22,e I .:L^c;., lr., Jf s: ls J-t Cli jl .5,- -f .\?
b) f(z) = sin la ; z=ni
d) f(z) - (z* \)','z=Yi
f) f(z) = Lta ; z= \
. sin hnz,) t
flz' cos h!
b)* ;2.=i
,r sin zd) e-vrr ; z'
a)f(z)=cost;t=tc) f(z) =zd i z=- \
e) f(z) = sin z' ; z-- o
.L-;3i *g: ar cJ-i orlr JatE ssU J:Clj jl 4,Jr irbJ la*, .tv
,{a) ,^y ; z'=l
. z*\*ic)@iZ.='t
QzcosL
4#
.f -rr., i,Lr+ irJJ,K, 6r jl c\-z
.e"a)7z'
q#
ltY
u.\A
u# o_s-a-
n=ln(n-\){'' ")#=":, nt''
iylr Fl, -2).e=Z lis= . )l cSj-t rl-u,.lrt * !Jr<, 6f :t ,S-gl-*l U .\ q
6-r+r,iL5 .,!F-^6
Ln(z+\) =,t= I
4#q#
. r SIf7fiZ ,c)y TctzItl:r /L
ffi*z'-Yz'+ \M
(z+ \) 1z' + \ r)
, ?-Yz/ra
-
"' zt +yz'
n -zr-YYz+Atl
-
J' z,-62' +yz
JI
|; I'lct
rlr -2r/ .6lii f(z)=z -t Ln(z+ \) tz* ' gl.rl q sf(.)= \ o6y' -r;S.iti .Y "
.:,-rl du-; ltl. \ ojy
gv ,rl;i 112.) =f ' (2.) -- ... = /^) (r.) = o t .r^lt ,J-tr z. ,trf .51 6 -rJ.-rti .Y \
f(z) ; z*2.
g(z) =
,rh7'-+') P') ; z=z'
..r-,1 sfLaz, gt+i d"$ JrLei.;r n 6li;l &; o.riL r-.,L, .:*,1 .-,r$, .VV
u*z*Y
e)
4Tlzl=t
9Tltl:t
fi Aly/,|-tll ;l .5r; r.-,,t.-r.l-,,1 .r;lL .YY
q{ nnzdzlzl=t "' f,,:','o"
*
d) ,!,=, *a'
n,!,=,ffi*dz
ilTlzl=r
OTlzl-t
OTlsl=t z'-&'+lz t)I
lzl,'r
d' cos la ,
-42
co' .Y
cos la..--- dzz'-Ya
cos la
-dZ
z(z' * \)
-zY-YYz+A
h) I t',::''nlzl-r filn Z
..r I t .zr-ylz+O,r
rt r',,
dz
ql'"6;*
f) I'"offi*ae
b1 1'"..& =o I +ACO&
rj.rtoltFt;t dy
u'i "#;ffir
,,{," ffi*
r.-rL. C.-rl .r. jrlL, .11
4{'"#
o,!'*#eil
s)!'*rrffiae
6.1+.1 0L:;.y0
4-{"#=,,'fi
DI'"1;ffi*ae h)!'"ffia0r
tl(t-a') ,'-\(a(l
-, ro de fiac)J @=W ''d)\
qlnsin'nodo= !!n)! =o,;
=W'r;n=l'Y'-'
Ji g;Vl.S)l;l .5re 1-"L' \:-"1 '-'j)Lt 'tf
4l-.Dt
4)CF
0! (x'+ 1,1(rr+1)
* \+rtt'n= fu\ +l'
,- cosxdxn)-!-@
ocos x
dx
f) I q [---
9T
(x'+ t)(x' +Y)'
o xsinxdxffi
ill-
u!
rt+ \rxt+rt
:--serg-frr+ \ltt+q
- sinlx .
-ClX
l'*x* \
i)i*ffi*o) !_
s€*ax
^t!*-ffi*
dx
dx
ilT
rf .t rl rlLs !- e't' 6!ss = 4 ttor[r:-,|9R-+oY.
E ,'o'Y
,it 5-R+ib eR+ib i b,-b e-itfr,rtJa:-. tlf csst J.r-" :l ,S.A|*lb.Yv
{-- ''" cos lbxflv-
!LA, cfi jr e J't.9t .
.r$r*,t-r lt jt c9u J.,rr$F*rl qb t dx(x+\)1xt +t)
*2, lt+ lr,'7,2, lt - l t,
*2, l> {r
;l-ti. .Y1
.f +., ,lL,l .Yq
+tP Jit-, ,A.f-+,f -"ttt.\
l')'l r' l)*ft1I t, l'* lr, l'+ I z,'+zr'l
-o
,- Lrut-to \*xt
dx =o
+ lz'l') ( r
= t( I z,
=(\-la)lz,b) lt
-21 It
'zt It
I'zr
c) | ,, lr, l+lz, l)ll;;Id)lz,l' + lz, l'- lz,'+zr' l s v I a, z,ia, 1, l' .
a r' +ar'
ar'+ar'*o , ,*A- rl-tel ar Jar di;l rS
zrtzr5z,bb.6l;ilz, l= lt, l= lt, l= \rz,*zr*zr=.o6r.r.y.Ji...r r(, oy-lr 4, tr tt-, Ct'eyl 651-.,;. d-L, 4. g-it
zytz1 ,z,ol(;ilz,l= lr, I =lz, I = lz, lsz,*zr*zr*zr=.o6re.t".sllJ. el* t- t o)r.,JJ.:*. ,Sa u,;rz, 3
6r;5c"U .Y
b) cos iz = cos la
d) cot iz = -i cot lu
a) sin iz - i sin la
c) tan i2 - i tan lu
.5+t* et*ltz y>W r-.o .0
b) ltan lal=1 a) ltan zl -- 1
: .r;5l.rg.:-.,1 o-r^i' ,.lQ ,)-i Cf e, +5 Jit Ji-r:.,,|t: b -l5Jt<-,:l .t
Lru 6 r5^.-r Ln(-z)=Lru rr*3 s> tlLn(-z)=lLru t+i;t s (z)' =z' ,-,-r5...:,-.,1 lru ),cl LU
lr., r, .t'r", -L-ys.s.,- t, Uffi , W, ft , + ct-r ;t .5yt.r5 .v
.r-5 ,J*rl {b,i, dr',| i 'r'' t4 C tJ 4 u. Jlii ut ..iLa;..
,-* u.J-iLi .rL.o.s.r lr* JrU ,6lrl =R orrlr j;lr f € 'l It l
"b .5-r+, iL:j 71
..:*,,1 F-"lF-.+t=t n=r- ) , w=z* ) o9l;,Ki -il 5--.r.o t: b I zl =\ oylr uli .q
-\r5-r5-ti.r-#rf -L-r;z=o f f,ql? ^-.-rw=e
z Ctt.\'
.c.--i tbyt lJS,- t-P + )s uX,Cl.t ii oylr gj Jtl, s, b
.r:-,,1 c-,'t-,lF- ,t+lorgrl so. {, o( I t I < \ q-r-i '-it6b(c..r;*iL dtJ^t n 6li;l &-.r^ 5+tr S.t*ltc sb sa .\ \
a) f(z) = x*ay*i(bx+q)
b) f(z)= cos x (cos hy + a sin hy) + i sin x (cos hy + b sin hy)
ts-l 2;fQ) = I t'-y' | -ltl ,yl Clr r5-rSl.rg JaL;, .e.;r 2tl2 5,lu-lii .\Y
.r-!b ',t-tJ
0,p,v,tt Cli )l ,-*-.tfy' -r5-f:..:-,1 ,rlJht e.Vf(z) = u*iv = peie Cb .\f
.J-:U ,*1Vf(z) Ctt.t.:-,,1 7:lf(z) tlr; ,r.$; c.{-? q .t*i "r,-,tr ,+ti
JrLi, Cri jl sj.tt.ht . ( vo. )
,3l-.'.rL.*i, La r, dr-,*r - F \|; tf\) = | 6 Ctr.Sr5.-.tli .\f.r:,.,* y-,F'1.i", &.ii dr/l I JS J,Sg.
vx t uv ir.rti:-:, .t<iT -r^!rr r-rr-r, y. v^ y I , w--f(z).61 .\0
.Uy=-V" l Jft-;r-f
r-!u sa-r,Y. I # I .r-r-ltu-i?&.1, sw=f(z)=u*ivgb.r, v tu.61 .12
.o,.,lyiri:*,2 ,L;i,tf1l vJ @.Kji-r-,tr r-r f M.
"rg Y J o),*/i+-.i, w=f(z)--u*iv srv ,n c61 .\V
..1-,1 gt Lt.:-,2 &it gfQ) Ctr.l<;i.t>Jrr;:liC H# = o o))eq fi.r>tJ eJrL^. .5-r+r dLr .\A
.r:.-,14Y,.lt; ;t---r .5h,#r u tSty.;L-vr Uj .t tLn lf(z)l ,f(z) orgf(z) , lf@ I ctr;t.{n't.rs.t<jiJ,-!u }gf(z).6rr .y.
..r.:-r slJ-t f d..htJ ,/J..t rlft l $tf(z)rS+k Ct*bf@=u*iv,JJ^rdby; J,3t-, jt &-1.2r .yt
a) u=d (x cos y - y siny)+tsinx sin hy + x'- lxyr+y
b) v= Ln (xY+y')+x-yy
+t1 "=*cJKr tr l.ryj A;!.;*.. jl 4t rJtir.yy
y=xt (d l=x*b (c x'+y'=6y 16 xt *yt =4Y 1o
: y-.; ,,liL jf 4 rr.2r .Yf
.+t* n= 4&r+ tr bY> o , x)c rgeli ,-#(a
.+tf n= l-1,-L.Jr-+ L.l2lnu>. 5 lzl <l ..-b '8
(b| +iz
( Y0\
.+t* w= " <0 <
:.5+k ,/y.y..1.,-+ .rr
.rrQ ,.)tr cgss !l: jVl .^;r d @
.rtQ ;U-; .^;e 6 cgss u,t: ;VY l,.;4 € (b
.r.rQ c*,1, i* .^ir 6 cgss q b jtit b;4 € ('
.r-11 .yb gtf- LSi o'.lt ;ill a,';e €n={ &' aK 'S-t'r: "lL:; 'YO
.r.lK./
,-)rr cgst u.l.r -r-ll ,f I lr.l < \ gtlw=da e,cit* eir'rr JL:i 'Yt
.r-16./
,lr-.ti..(.,- dJ=,rroLg.r:rVg 6t+ bz. c.rtj dti; l{ P;r r[-ry 6- 'YVu-ott' +'. Jljg\ 6,5.a-"
\ = \ +h iz.*qw-z' z'z' .\>tiw=z*h ; z.=a \
..f,-r+ rrtjr.lr \* = * +h 6t'P..1.,-t* g'-t
"+t* ls
'l'rls> 'YA
.l;:-,r,1
.utsi c-,,) ttw=t rl',-t+ ! l.ry=x+ | , y=x , l=b*b b-F :l &--* Lri.,,lj 'Yq
:+L,t w=arc sin z cit<J tr tt y*; ,flijl 6- y' !r:'ii 'f "
jVt bJr € (o
.J-rU cJ-i rJ.,y x yzv t €ls(\,o) , (-*,'\l J* I+Sz A-ia(b
J5t g2 k
:+lk o$ olr cr!,Ji tr tt.r*1 ,flijl .4, Jr LJin 'f \
w= cos la 1w= cos z l.y=c tx=c btL; (a
L r--V ,skf''Jfr "L'+'r',
w=coszvy>o.-Jt <J< o (b'rJyy\-w= cos la U x) o , "<ycn (c
w= tan z try> o , o<<n (dw= cot la I " <y<n (e
.li;r rSJ lzl e r-,1*, r:,-,1 ,-,.Jt .fyc
z=-i,Jof, b .rL".c,;., lr., ;l -F o2\c (a
z=-i rtjt;l Urq- + sargzs !-, Fl=\ oy-t-r-; Jp 9c (b
ltl: \ oy_l-r-^; J-U *-: ,;-.-,c ,.li -rr O f ,lzlZ
fu +6*:*,1 .7jt .y1,
.g-,,1-\Sr<\ 9/)"
.-t;Sr*,l.-, Jr -,# tsu Jls f, 1at JrJ:r .rr
:-r;5r--,,t-., Jj tSb.*jl 4, -f J-p I b I , +Jfrftf .fO
,I l ., ltl=\ lyl.r-; Jp g @
lzl: \ iyl: J* t'(b
-rj cg6.*jl 6- -f J* ttlt I Lru dz r.-"1-., .:.-,1 ,-,jlL. .Y?c
l-l.r iy_l: J-/' , (a
Lra = LnR + llti e-f \:lt | =R iy_l: J-rb i@
,.:,-,1 13;)Jt n tsi g ^S I * Lru fu +* r:-,1 .-r;J.t .yy
lZl=r
.+sr-,t-.r -tb'.r.1.-,1 ,S)bc,.1i.,,, ,6lt I z' dz JlFllzl= t
ynR lol *R tsljl q rf -rySc.:ti
.rA
.rqIlzl=n
ldzllr-o I lz+al lR'-lal'l
.t<ii -r-ltr *-r* .rt.o:;., it r ;l dt --^ &-:rf(z) otfr.r .f "
lim I'" flrr 41 ag= vtf (.)f+o o
J-lt; c ,t gl9z..l rl! JJ-c r:-ro)L)y 6Jtl irl.rr lf -516rrn, iL5 .f \
d f'Q)az= d fQ)=* 't*i', z'2. 'c Q-2. )'
JlFl r..,,1^, r:*,1 -,r;I.b, .YY
zd dz
ffi r-,1-r.:-,1 ,-rjlh. .ff
.!:U c $*). p
lr.r, r:,-,1 oJ^!.1 ),et C orL rt.-r. jJ.4irJ e, +5;l.rUlr5.:-b ,5.-2tf(z) Cb .ff
lm :lr;Li -f 6ljlq.t$c{tt.-uq ,J ,Lb g-,lirT.rUt, &n dr- c.'t al
I +,.4)\lz-al=a z '|
Fl, rr Cl.l a tLi; 6 uttt'.,,r $ f,
!Li. eti:s aAt.*r..
\
lltt
(Yor)
(z)hrui2-f(2.)-f(")
..:-,1 6$l-3-rl Ur ,h;iz..li;.r 6
6 uttt'lr1} f, z' ln + dz Jlfu;l .-L,.:,-,,1 .-,1,L .16
..:,-.,lltl=l cyltc (a
.J-!Lz.-\+i ,s;Jl.*lUr.LaJor; lz-\ l=\ o./lrc (b
-F s ory ,LL.;D .r-ti lf\)tc c.rti eb J$-r Ctt.4 -Sl 5.r5..-rU .l?
.Srt ;l .sly;;D J-lr.rr b rr.rl& f.# lf(z) I .ttui tlbq3b rtrJr.t*i lf@l s \ sf(.)=o tory..rg; lrl<\ 2:fQ) 06y' .Yv
...r*.,1 lyy f i:i-t- d q ^+.# al .lf(z) l s |tl
I 6 f(z)=e i* .t<ji lf@ I = Ir I 'J-lr
+ti; 4,.sbt \ -Slr5-r;S,*u o:+-^..:-,,1 &d)JeAtST
.J.y -rtSt lr g--t'U ,1,-r! Pl:r:r.r il ,f u s^f ,r+Cb. ,/bbil s$l-S;il;l 4, J^ 1?,-L, .:*.,1 *rrlt .Yl
a)rdz1,,=,
\Yz-l1dz c) I {ez dz
lzl= t
frc
b)llzl= t
sin' \ dzz
l_\z +ez'\ +ed) I (\+z+zt)(e
lzl=r
flt-tulZl=r*' "'Ei
dze)!lzl=o
dz
(\ - cos z)sin z
9T cM iC:yt =Y
fi ,S!Jl-&rl rl 5-t l r-"' n ..':*'l '-'-,'l.t-. 'Yq
al fro cost yf.dQ i ; a*t \ , aq-.(' J o \'la cosQ +a'
b) i." ec' o cos ( n Q - sin O ) d O ; n€*N
,tc) t tun $+in)dr ; ae.R
d) I '" cot (x*a)dr ; ae( , Ima* o
orf- xcosxdx Df =".*,t.fug J_ ffi t. -6 xt+Yx+Y.
. ,* xnx')!- lffi r)f &;n*N
,- sinx dx*rJ, ,t f- x'-b' sin ax
dx't J Xr+b, X_t- (x'+t)(x-\)
,JldlFl r-,,L, u.il r.' 6st i zP'' e'- eE :t g.At-J-,; tl 'O'
a)o, o ( p< \ , t" xP'' cosaxdx
@
alo,-\<p< \ , I xP'' sinaxdx
T.
JrLA. eti :t sAL*r ( tor )
.J, ll.r -,< Jt.
JfFl i-.,t^, \ tt F cgst J I h(+-4 + r-.,tr.rq .o\' z \*z
h(+.4#.-t;ll;
.!5c,ti .0Y
lr-t I lzl -t I + lzll aryzl (a
< \ o6,abYJ
lr, I o6re (c
.+S-rU .01
l(t+t1z'+izl<
"rs#=+ "rs|.G3ilz,
f .r,i;'1l=lz,l=
( YOv r......
a) (,/Y-i)"
b) ( t +i)"
c) (l*cos a
u(x,y) -
= y'@'+-isinffl= lntr @r+ * urn ffl
+i sin a )n = rn cos" + kosff
( ,; lyl > lrl{[ -, ; lyl s I'l
u*,-;:-:., .Sbl, .li -2, )S ottc.,La:;., f lf.r r, # , # ;r.i1fu., .Sl.rl,
.c-+.-tri"r lsf ' (") 3r:.-,1 ";*, 6,1'-rlsz=" lWfQ) = z Rez Cbd-r.+.r,.lLi .00
.+t t w=z*d.1,-+ \lt-fr S y S tt +b ,-Li) .6,
J-.or rjL'lj .0V
\y'' = cos (lluc r/l)+t sin (llctt r/l)
.+t*' ltlog ,l)-t)
, ,oB ,jt
*tt .t<ji tog* = #rr*s-;,-r^, o6r,r .0A
no+ rstn ;ldtr r5$r iL5 .0f
+f-r+, JL:; .0q
UI
ql-!,1-dx-,o i o<a<\. \ *.r sln afi
or__-\ ou ou _ \ ay0r r A0' Ar rA0
€ hx ' ln'/T+bT b
@ca = b 47s14n-
.b-l,l i J-l J-2j,.,a +. u.bt.it.a;. -l) dr-__l ,g.iY.lL.. 6-t-.or JL:j .t.
laLi, Cli;t 6At*t . .. ( YoA )
.t*i r-:,t ,rLLr e.Vf(z) o6re .2 \
(#.# ) tf @ t'= r tf '@ I'6-t r.r jL5 .tY
4! sinlo do- = + [ -a,+ | ob, *(a\-ilfl 1
a *b cos| b" Y
cos'n 0 d 0 - \xfx0x..(Yn-\)YxYxtx..x ln
lt ;nY
lt
U [,o
CN
.$J- lJ r-.2 rFL.ll .5rr .Pf
a)V'u=x'-y' ; o<x<a, "!1aur(a,y)--o , uy(o,l)=o , uy(\o)=o , uy(x,a)="
b) ut = c'Y'u+ry sin t ; o<x<lt , "!1fi , t) o
uftr!,")= x * y
u("rl,t) = u(n,yrt) =" , u(xro,t) = u(4n,t) = "
c) ur - ct Vt lt--o I o1lc(ntr, "1!1rt, t7o
u(x,y, o)= o, ll( "ry,l)= o, u(n,yrt)= o
u(x, ",t)=l (x - n)sin t, u(x,n,t)= "
tt &lrht d:"- .f- t:-'"'l -"tlh' '71
Y'u = f (x,y),' V'u= V' (V'
"(+,t)="(+,t)=o,
Tru 0'u |'tt_ f v ..L
-ox' I I ax' ay' oy'aart,Y)=us(1',1)="
u)=
uo(
(YOt).... ...... t..... ..,... .....r .'.'g.ltter[-ni
u (x, o) = u (xrb)=., uw (x,.)= uw @,b)=,
..tJJ- .r>t) cr$-q.SJo lt ti Jilt -, ;l S.tt .?6
4# +Yr #=rt; u(\.)=\,u(,,t)=\
U'# + #=o ;u(\')=' i x7c ' u("t)=o i t2'
.JJJ- b.r.^:.. .s:<,it ')y \b li Jt-, il .5-.2l .??
a)u;uo*au=" ; c1x1nt 1t>."
u(.,t)=uxQt,t)=" , u(\")=x(n-x) ; oSxgrt
b)uo*ur=" ; "1x11t, o9(\
u(tc,")- u (l\)=sln'x , u(",!)=sittl, u(n,y)= "
c) uo*ur*ut=o ; c<;1tt, c!(lt
u(x,")=, y (16t1)=" , tt(o,y)=" u(n,y)=sinf
d) uo* * "r* * u* =.,' r(\, .<0< L
tt(r,o)= o ,ue(f,*tr=. , u(\,0)=0
e) uu*llt;c\4o=o ; o1x11t , l7o
u (",t) = u (tt,t)= o, u (x,o)= o ut(4")=x
fl uu*u;ctlto=66(lx1 ,' o (.f,( \ I 17 "
u(.,t) E o, ux(\,r)=. , u(xr.)=sin** , ul|o)=otr t I
g) uo*uo=sin x' sin' x ;, o. +, . (.P(Y
u(.,y)=. , ur(1,il=" , ur(x,o)=o,uy(x,Y)=o
JrLi. eti :t e ,t.&r;t . ( Yt. )
(Lr.=:ir .55,b ,-)s: q) JJJ- l, y; j"L ;l 6-rt .?V
a) V' It--c i "1x1n , o1f 1 It , o1z1Jt
y rll ,u,1lr1z=trt t y=rt t y=c I X=lt tx=" 4iljl u. u )lJi)u(x,y, ")= sin x sinr y
b)V'u-u=c i o<x<rt, "<y<+, o(Z(\
It= o fslsz= \ J.I= o tx=. gljl t
SudurE6\ul=o*&l,o)--lx-lt, ,r rr-_t ,E | ,=n
c)Vttt=c r'r(\ , o1Z1 rt
u(r,0,o)= u (r,0,n)--" , u (\,0,2)= z(rt-z)cos' 0
d)V'It=o,'r(\ , o(.Z1 trt
u(rrQro)= o, u, (r,O,n)= o, u (\,0,2)= zcosre
5r+r.j[-:j.Kii .t-ilz--e i0 ; -tt <0 < fi oylsc olSg.?h
-az
I e 4t:1nicz
l5-r-.u Jt:;0 ..-.- J j-rt JlFl ,Ft ! .c.-,,1 c+ti.l ,*-,f- 6)b a ,si 2: $
I Pe *os cos(a sin\) d0 = n
dtl^t ii ptr f 1tu,-tD ,lt;lfJ q q-tr .t:f\)=u*iv Ctr rf+t ,-i-f .?\
u(x,y) Ctt r5-r=l: .rL: e f@ eV V ,..,JU ,f..ol .r:;.rtSq t: .t*i .r-tL -U.r: .r
.J.56. -rt+tl D Olf 6,U s )f 1'-;5L. JU.A.
(url:.rjf) ^5r+.l .lLr .Vo
( Yt \ ) .... .. Crtea Jrt
2_
I' ,'R sbro do< + ,. R) o
a) {- sin'n0d0=ffirS-t-.rr jL.l; .V \
.Sr.n, db y*.1 );t cgst I e -JJt-ti U .VY
b) {- Ydx=L
1T
\ +/ n sin (nln)
lLi. Cfi jt e$yht (YtY)
€tt,
I. A first course in partial Differential Equations , Hans F. Weinbergar,
XEROX college publishing , 1965.
2. Advenced Engineering Mathematics, Envin kreyszing,sixth Edition, Iohnwilcy , 1988.
3. Panial Differential Equatiaru of Mathematbal physics , second Edirion ,
Tyn Mint - U , Nonh Holland, 1973.
4- Advanced Calculus, second Edition , wilfred kaplan, Addison-wesley
publishing 1973.
5- Mathematics for Engineering students, P.D.S Verma, katyani publisher,
1977.
6- Higher Engineering Mathematics, B.S. Grewal lcanahan pubtisher, IggJ.
7. Functioru of complex veriables, philip FranHin, pitma4 lgsg.
8- Introductory complex Analysi^r, Richard A. silverman, prentice-Hall, Inc. ,
1967.
9. The theory of Functions of a complex variable, George rankovsky, Mirpublisher, I97l .
10- problems in the theory of Functioru of a compler variable, Vrctor shiffer,
Mir publisher, 1972.
11- Complex Variables and Apptications , third Edition, Ruel V. cherchil,
James w. Brown and Roger F. verhey, Mc Graw-Hill, 1974.
12. Elements of partial Differential Equations, Ian sneddon, International
student Edition, 1957.
13. Introduction to partiol Differential Equations with Applications , E.C.
Zachmanoglou and Dale W. Troe, the Witliams artd Wilt<rhgs co , 1976.
( vrr ).... ... pq? J.ei
I4- Introduction topanial Differential Equations , karl E. Gusrafson, fohnWily, 1980.
15- panial Differential Equations, E.T. copson, Cambridge (Iniversity press,
1976.
16. patial Differential Equatiaru, P. Duchateau, D.lA zachmanry schaum,s
serics , 1986.
I7- Advanced Engineering Mathematics, M.D. Greenberg, prentice HoltIntemational, Inc., I g8B.
,tu-:g e-#.r\rydWt*U-: 9.f >! -t* .l at-.1 I,g -*.f*,J+
u-l 9 r-s- -# .r\Li . J.ll J.l-rt
f(x) i, @) = F,(f)
\
Y
r
f
0
?
v
A
q
\"
\\
\Y
rI \; c (x1a
1L o,' Fr..ri^t-
to-'; o (c( \
-axe Q)o
e-x\lt'
, -*'; a) o
{ e'*
l rotn c.-x1a{
L ',. A,.rt t-
cos ax'; a) o
sin ax'; a) c
sin ax i a)"x
e-x sinxxt
Jo (u) alo
,ft sin awttw
,F 9*, onItw"Y
{; (*te 'wr lt
-J: , -wr lw,( la
'FA i#Re@+iw)n+'
# try.wI
,/Ta(a-w)Y"
I orr ron l={ln w'
{t H( a,- w_ )lt ttl a'-w'
*,o,(# +)#'o'(#.+)
e-;Jg-9i.*,',tlA,ry' f l J.l-fa
\
Y
r
f
0
?
V
A
q
\"
\\
I \Y
o <X<A
,4r.rLl+
c(4(\
A)c
A)o
A)o
[r t{L.;
\16
\l xYl'
xo-r;
e-';
,-* ,x
{ ,'* ;
xz'xrlr
o-*'i a)o(I snx; c1x1a
1L " ,' -&lcb,4
COS (Ni A)o
faArCtAn + ;A) o
x
f' (w) = F,A)
qtryl
Y/;
,FJT
,/11t
,/1JT
,lErt-w
we
ltu) altsu7
-wuY
(*)
arc Mn La
d:#tm@+iw)n+\,1,
w -w\ lta
-o
(lalYtt "
\ [ sin a(\-w) - sin a (\+w) I-
Tl,llnt \-rry \fw
H(w-a)
sin h aw -aw
-e
w
lrbi, eti :t SJ,t.$t ..... (t?t ,
u-tg sY{lt'.ff Jf.t+
f(x) iro = F(f)
\
Y
r
f
0
?
V
A
t\.
{:,
{:;\
x'+at '
{.it"
{:-
{.:*'
{ :'*
{ :'*-axle;
-b<x<b
-4, abt+
b<x<c
.4r.-rr.h
Alo
.( X1bD< .r <Yb
,i, .ri.t+
X) c
,.(1> gtri;
b<x<c
-&r.rLl+
-b<x< b
-ri,.sLb
b<x<c
,.(1r.fLut
A)o
Alostn (a
,x
t/E sin Qv,tw
,'b' - r'b*----r*!:_iw tlltt
- rfr , 'elwlY--Ya
-\+reibn -r-ribw
-
t{ltt w'
\-rr+,l ln (a+iw)
,(a-iw)c -r(e-iw)b
-
,,/ln (a-iw)
lflt
sin b (w-alw-a
i eib(o-w) _ric(a-w1:! ln a-w
\ -rvYar++- e{la
ftt"l<a;,if l,l >a
( Yrv )...... " " trtr.t Jl'ai
e=, Y.V \ AYA,/e= \.PYAVYe'= V.fAq. O
!!= f.\Y\0q,t1= q.AtqP.,/N= \.VVYYO
logr.tt=..fqv\Ylnn= \. \ YYVYlogr.e=..YfYYq/n\.- Y.f.YOA
y'y= \.r\YY\VY= \. YoqqYr'r- \.vrY.o'y'Y= \. rf Y YYlnl= ".tqf\YlnY= \. . qAt \
sU t)Lir Lf .il\AYAf 0q.Y0 Yrort\YV.V ..\YA \YtAOt.q^q r.t0. tYvYr
Yt0r0 Aqvqr YrAYtYY.\. AqroA t\AAYrA0.q.00\t oYvln
y=, ".oVVY \ dttf q
ln y=' -..01q0f qf \ Yq(cf.p.Y\f)
\ o= ".. \vYo rYqY0
\fd= Ov.YqOvv qO\f'= OV "\V ,YY. A.?,
qf \rr A0fr0YlY.. \VY\Y.rYo \ AYV'oqY.YO
'AY.YVf.q0 .YAA.qY^VT \'YVVTAAVV Yqr0r"VY.A TAYTT0qqY0 r"qYYlA\"1 tq\f.. \0rY A?.?\A\tfY AYYrf
\ qqYf YlOvvl'AYf Y. AVIA."
qAVY'qAA0A
YfA\q0.qYq
r0r Yr\.YqA.A.V0qov.rV\4.0Y YAA'
,#sL.d.i,x=rcos0 r=r/ (r'+y')
l= rsin e 0= arc ton l.dxdy=rdrd0
rll.#r csls.r dll*= t { llrl<\)| * fll-c
:fll= c
cosx=fll= o
@FT
ln(\-Y)= -Zl/l= c
@srarc lan x= Z
fll- o
(-\)^x"n,(rm)!
l" ( lrl.t/m
{m!
/
rlrl<t4tJ>dvt-
alphaBetaGomntaDeltaEpsilonzetaEtaThetaIotaKappa
Lambda
Mu
o,pYrld,ateq0d,oL
K
vNufxio omicronrt pip Rhootr, sigmat Taau,Y Upsilon
Q,g,Q phiX Chi
9,q' psi
a,Q omega
)r,L
p
Lr.1l.:y
]"'1,,'i,rl
a.b =a,0,* arb,
axb =
curlV=YxV:
srad I =v[= #,,* ffi* {odiv v=Yf= #. W. #
a a a
At Ay A,
b, bYb,
JrLi, Cfr jl SA|$t ... ( vr,r I
(cu)'=cu' (c corctant)
(u+u)':tt'+7)'
(uu)':u'7)+u'u
* l'- u'u:l'u
du du dv*:- = +. 7 (chain rule)dx dy dx'
LSWfI ,lJ t.5,r:)rlEt LSWY.f €J.
= *;r @ cos bx*b sin bx)+c
Iur' ly - u, - !r' u dx
[l a, = ll.\ *c (n#-\/' n* \
,f + la - tnlxl+c
f,*d,= Lr*+,-a
fsinxdx--cosx*c
[totxdx=sinxtc
Itonxdx=-lnlcosrl+c
[ ,ot x dx = lnlsin-r | +c
! trc x dx : lnlsec x*tanx | +c
[ ,tt x dx = tnlcsc x-cot; | +c
Ihtu=jorrtonL+,I ffi=arcsinL+cf .r,{ o, = sinh-' jL +c- V (x'.*a') a
Iffi--cosh-'**'f sin' x dx= \ r- I ,. Yr*c'Yff ,or' x dx= I r+ I sln Yr*cJYY
I nn' x dx= tan x - x*c
t ,ot' x dx= -cot x - x*c
fntdx=xlnx-r*cI, * sin bx dx
e*= fu (a sin bx-b cos bx)*c
fr* cos bx dx
e*
({)' = nxn''
(t)' = {(t)' - ttn a
(sin x)' = cos x
(cos x)' = -sin x
(tan x)' = seclx
(cot x)' = -cscrx
(sinh x)' = cosh x
(cosh x)' = sinh x
(ln x)' = Jx(logo x)' =
lo4o ex
(arcsinx)'= #(arc cos x)' =- rarb(arc tan x)' = =L' ' \*xT
(arc cot x)' =- +:\ +r'
.e.r*- t 1.--::, dt4\ry
The most imponant systems of units are shown in the table below. The Mks System
is also Kttown as the Intemational System of Units (abbreviated Sl System), and the
abbrevi^ations s (instead of sec) and N (instead of nt) are also used
1 inch (in.) = 2.540A0 51 cnt I foot (ft): 12 in. : 30.48006 12 cm
l yard (yd) = 3ft = 91.44018 36cm I statute mile(mi) : 5280 ft :1.(fl9335 km
1 nautical mile = 6080.2ft = 1.8532 lon
I acre - 4840 yd 2 1046773m 2
1 ftuid ounce = 29.5737 cm 3
1 mi 2 = 640 acres = 2.58999 87 kn 2
1 U.S. gallon = 4 quans (ltq) = I pints (ltq) -- 128 It oz = 3785-432 cm t
1 Bitish Imperial and Canadian gallon = 1.20094 U.S. gallons = 4546.1 cm t
1 slug = 14.59390 kg
1 pound (lb) = 4.448444 nt I newton (nt) : 10 s dynes
1 Bitish thermal unit(Btu):1054.8 joules I joule = 107 erYs
1 caloie (cal) = 4.1840 joules
I kilowntt-hour (kvh) = 3413 Btu = 3.6.106 joules
t horsepower (hp) = 2545 Brulh -- 178.2 callsec = 0.74570 kW
1 kilowatt (kW) = 1000 watts = 3413 Btulh = 238.9 callsec
oF - "C. 1.8 + 32 1 = 60' = 3600' = 0.01745 radian
for furtha dcuils see.e.gD. Halliday an^d RRestick Physics.3rd ed. Nar York Wilq.I978.See also
Atl American Narional Standard ASTM / IEEE Standard Meuic Practice. Institute of Electrical and
Elecnonics Engineers. htc.. 345 East 47th Stcet Nav York N.Y. 10017.
System of unils Length Mass Time Force
Cgs system centimerer(cm) g'am(gm) second(sec) dyne
Mks system meter(m) kilogrant(kg) second(sec) newton(nt)
Engineeing system foot (ft) slug second(sec) pound(lb)
L.3l; r>tp
"L-t+\vr dr5uL;\ vY ,rJa;;r
Yn U/JS,, {r-2;i
\ Vl g/JFJJ,
Y plJ +!L;: .t,L;-lrr .1 :ri t=n;
cri\ 0t ,rLr-;
\ 0f .\ T q .hi:i,\p, 1[_-r CJrt'
\ i0 ;Lri,\ l. jL_-r
Y \. JJii;
J.l-tr\?0 +-19 gr-;+
l?? +-;g
l?l u-1g,-r-#\ \ Y ,st! 6td.+ r+
AV CJ, d.rL, .p)l: ;-\A\ dldU L;
AV r*')l'r\ YY dS,-r,-r
\v .\Y ,JJ - ar.-,-r
,-tStyt ,l.r-
VT Lr +:, .tS.t-
6J-'\. rr_.lJg vr-F
Yf et51r r1;y
1.Jb qry!. r,-)J d#
Y\r jr/'.
t' I F'*r-,,5i..cCi..;-l\ eQ lai:'i,. rli.cl
\qr i.li, b;4.p L) 6:: jtSsttt,4-
_.t{J',-5:l
Yt'JiYq .YV .\ r,-;;l
\ qr Llii. C'r jl 6;j1&ld't<'.';t
\V\ glu-iry\ 6ioyl:
-L 'tJ--
r -l:j
\ t V d:t^,\t^ Ji/.
\\q.\\rJ-.,v" bT u;..
0 \ J|J,-.L\ "A .,r-l*
f" .;l:i\00 Li.:i, gtr F-t*
P0 .\ Y dl.i- r-l;-r-z
cu\ \C fl-, qr F
\?? "tb;
\ A" Lra\ P0 jL^
.)\+\Y" .YA.Yl .\ r,;1j
fV .fP ,r-.r4 S "r,-2-r3
r--tg
Y? oljs-. 'rr-r+ t e-# qt\ Vf ,.aL- 6-J
\ Y. .J"\)
t .e! C-r4.1
I\^dLJP0 ;;:'+
f 'n .|'-l:JI \A ^,lj r.riL
LLi,\ 0r gt;\ YI rl.tel
VY "511J't..
VY glef1 +J1lr,-rtirt ,.lit"-yo rJ...
\ \r\ ;,J:*t.=-U.i- .,")f'lt\ o o i'iti..l jJ-
'y ..1=a .(-;r .:21.; J'ui:lqY tf
\ \ 0 'f .-t*- gl,t .r"\)Yl r.2'(:
.r):L^'?\ if .:,,i.Lj, L
tq 4r.r dl-t\ OA .\ C? -r--.t s)S
rJ:i.^-
\ \v irr\\q.\\tJ*V. oJ r;*
v' ..rlJiv' JJ,-r+tq .rld .-:qt .v. uJ
\.A .V. .rA!I "1 g-lf .:L-a:;,:, .r\Y
\\V.\\Yjt.l!Y'cy
\.t .5-t+ 1r ,5La.i )t Cyv. j-J.-Ji*^
\ty .y. .l-ra-I^Y \ r irJjs,
\ O \ rJiJ,Y. q J5 J.rl U
s4-rY\';l;
y \. jtJ, s ,-#\ qlt
v. jJ,-#\ 4,. ,r-i1t(J ,l"at r.-tJ
\ . n ;pbt-.. .:rt'-'it "Y \ 0 ,r'l'l"-,' 6b 6Lrrr..,
ld:i, ost'Yl r,-2Y JlFt
Y \ +,-;Y 6-Yq 'f *-:l --l.e
vr2 "|,7
\ O \ rJi :, J-rtY Y Y jtrJ \ ;,-t c.* i
#Y.Y g-,-fL,Fot
Y'q J.ry lrlet\ lv ,r:'JlFl
Y \ \ JJ'L;
\v.\r JJ;r-,,Y.f ,r,:5JlF"l J-rt
\OV .\ O? -r--2 f\ 1rr ,r-t'Y \ t ,-,lJ:J
f " rr ,f,_-r::J
Y \ 1 lro$L
:-L;YTY O:L
\1r 7m ';Y\ OV .\ O? V-: uJ
t 1,r ;-J,;;3
1 6rj1
'-P\.A .Vo ,.f\)
\ \Y rJll
L;rl;C-* . (fVV)
, L;:
YY \ J5;YYY ,r-Ll j^*
YYY FirJ J.i\l\.gl+;l-:
YY\: j:,.gilgj
\10 .\0f ,-,-.t-,r\y7 W=ez
1yy W= sin z
\v. 17= \
\?\ w=|t12y W=z*b
\ 0f t!i, ab y ,r-l;Y" i-S Jii.
-tt-''t*"\ . q rjld-.i\ "n e..i-.a
Y. 5l*(,--ir-r1yy ,y" F,x-:_