Σημειώσεις Διακριτών Μαθηματικών
DESCRIPTION
Οι σημειώσεις μου από το μάθημα των διακριτών μαθηματικών. Ελπίζω να βγάλετε άκρη... Σύνολα, σχέσεις, αρχή του περιστερώνα, μαθηματική επαγωγή, προτασιακή λογική, συνδυαστική, γεωμετρική πρόοδος, ακολουθίες, διωνυμικό ανάπτυγμα, γεννήτριες και αναδρομικές σχέσεις.TRANSCRIPT
~____ +-____~~~~~~~~~ __~~OamiddotAn~~=u-_______________ ~~____~____~-~~~ ________7~_~fl~~p-_____________________
~---L--------------------------~-~----~ 1
7
4 5 - - =- ~6 ampA ~
-8 c
bull Ql1Y~
bull
0
)
rshy
~q
1shy0 () 1-1 1
0 1 0
1 1shy
p ~gtq I
0
~
I
1
o
JA x61= fAl-Il)
IAU~ f lA J t IBf~ IAn B1 - t9t ~fI t va E ~r~IG---___~
A~middotr 1Jrca-10~
hArol~ 10 TI 8~amp~ nxl)
U Un-i~)~o~__
tmiddot bull bull
bull 1
~~ ~
If~il~t -- ~-~ -=I ~of~4j sc J
4 ~ I ol4rlt~ 10 ~
~ -=+1 __ -~ =shy 9~Q 0 0
3 1 bull
flgt ~ o)tl91dmiddot cr~I(~~ 0Ift8l~ fA to1oll it JtfV~J ftshy
-atCf 0) -lOt q~t~t~
o
~ 4O 8 7[ IfJo --shy
pc ~1Qltmiddote1~~ fit It
~1fo1~ Jdt1( t ~) 00 t (- gtj itDI It ~II) 16 it bull bull bull (4 shy
gt( t ( 1middot 10-1 --gt jsmiddot 3 90c ~ -= 3 (lttomiddot ~
~
AAlClt~I~ AvtIlAEvw middot (Awfffvt rt or~t~ 0foifIJV oitlUflff(IIIY)
1f()~~r)JrCl l1ot~ 6u (1~0i1~P~ P1~ 5) fl~ -r~ylrq7~ 11) KAK A-O~~rgt
~ ~ C( ~~ 2~1
f ~ ~----- ~
(~~
l tv UJ 4 ~
4tlo1Q Ill (
~
1 d bull
81 5middot 3
bull bull Af)C1 33j middot~tJ 1~~ A S (jAto
22 ii 0 qDr f Clw Itl) 0 ~-~ 13 5 - 1(bull
G 4 -1shy
~----- 91 ftc
~ 0 -lIshy
t -shy
1 2zl __--shy
bullbull f
10 dfl 0 lt0rotl ~gt TJOfio fiJ1J
~J t J 1lt10 ) 10~ ~ J
10 ~ I
caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
UUrU6tei -= ~ttl 6it Cf ~t ~~) e - f a bull
bull bull amp bull bull -
-shy
I bull
~
a 0
~=
I bull
bull 1 bull I
((1)= 6 IIbull n- 1 r
~ ~~-- - I
C (7 - rmiddotL ~
4 3 J
t o
middotmiddott
r~ 5 1 middot ( 5
- ~ ~ 1 1_ t 0
1T~Q~uIf d~) Il ~1 11 ew ~flgt - rlon) src~ f7f~t(J
61 oflc I t ri1PfW 1tJ)~-1ln~ 6YQl ~ 7 bull )
A V~ J3J middot A c 1~ (Hf1-S It _)7 PI1 (1(f ~ 4 ()-1 I D B (1 f) - 1l f J~o ~~
aOtI4Jfft- Z1 oil bI1 )gt () 8 ~rl A- ((~) 11 ~III7JnlJ 0
1~ - -) rtlffinr(JII~ 0 J J _ C()- 8 fIf IA(A(
b ~
bull It~) ( (1p 11 h ~ I4f iJ 171(Imiddotf~J --C--C-1 1- S) c lt- - - - ---~-~ )tI o~i4 11011 5 t1j1Pf7j~h J~ al 4i~ C ( 9 5) 12 -
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
~---L--------------------------~-~----~ 1
7
4 5 - - =- ~6 ampA ~
-8 c
bull Ql1Y~
bull
0
)
rshy
~q
1shy0 () 1-1 1
0 1 0
1 1shy
p ~gtq I
0
~
I
1
o
JA x61= fAl-Il)
IAU~ f lA J t IBf~ IAn B1 - t9t ~fI t va E ~r~IG---___~
A~middotr 1Jrca-10~
hArol~ 10 TI 8~amp~ nxl)
U Un-i~)~o~__
tmiddot bull bull
bull 1
~~ ~
If~il~t -- ~-~ -=I ~of~4j sc J
4 ~ I ol4rlt~ 10 ~
~ -=+1 __ -~ =shy 9~Q 0 0
3 1 bull
flgt ~ o)tl91dmiddot cr~I(~~ 0Ift8l~ fA to1oll it JtfV~J ftshy
-atCf 0) -lOt q~t~t~
o
~ 4O 8 7[ IfJo --shy
pc ~1Qltmiddote1~~ fit It
~1fo1~ Jdt1( t ~) 00 t (- gtj itDI It ~II) 16 it bull bull bull (4 shy
gt( t ( 1middot 10-1 --gt jsmiddot 3 90c ~ -= 3 (lttomiddot ~
~
AAlClt~I~ AvtIlAEvw middot (Awfffvt rt or~t~ 0foifIJV oitlUflff(IIIY)
1f()~~r)JrCl l1ot~ 6u (1~0i1~P~ P1~ 5) fl~ -r~ylrq7~ 11) KAK A-O~~rgt
~ ~ C( ~~ 2~1
f ~ ~----- ~
(~~
l tv UJ 4 ~
4tlo1Q Ill (
~
1 d bull
81 5middot 3
bull bull Af)C1 33j middot~tJ 1~~ A S (jAto
22 ii 0 qDr f Clw Itl) 0 ~-~ 13 5 - 1(bull
G 4 -1shy
~----- 91 ftc
~ 0 -lIshy
t -shy
1 2zl __--shy
bullbull f
10 dfl 0 lt0rotl ~gt TJOfio fiJ1J
~J t J 1lt10 ) 10~ ~ J
10 ~ I
caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
UUrU6tei -= ~ttl 6it Cf ~t ~~) e - f a bull
bull bull amp bull bull -
-shy
I bull
~
a 0
~=
I bull
bull 1 bull I
((1)= 6 IIbull n- 1 r
~ ~~-- - I
C (7 - rmiddotL ~
4 3 J
t o
middotmiddott
r~ 5 1 middot ( 5
- ~ ~ 1 1_ t 0
1T~Q~uIf d~) Il ~1 11 ew ~flgt - rlon) src~ f7f~t(J
61 oflc I t ri1PfW 1tJ)~-1ln~ 6YQl ~ 7 bull )
A V~ J3J middot A c 1~ (Hf1-S It _)7 PI1 (1(f ~ 4 ()-1 I D B (1 f) - 1l f J~o ~~
aOtI4Jfft- Z1 oil bI1 )gt () 8 ~rl A- ((~) 11 ~III7JnlJ 0
1~ - -) rtlffinr(JII~ 0 J J _ C()- 8 fIf IA(A(
b ~
bull It~) ( (1p 11 h ~ I4f iJ 171(Imiddotf~J --C--C-1 1- S) c lt- - - - ---~-~ )tI o~i4 11011 5 t1j1Pf7j~h J~ al 4i~ C ( 9 5) 12 -
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
7
4 5 - - =- ~6 ampA ~
-8 c
bull Ql1Y~
bull
0
)
rshy
~q
1shy0 () 1-1 1
0 1 0
1 1shy
p ~gtq I
0
~
I
1
o
JA x61= fAl-Il)
IAU~ f lA J t IBf~ IAn B1 - t9t ~fI t va E ~r~IG---___~
A~middotr 1Jrca-10~
hArol~ 10 TI 8~amp~ nxl)
U Un-i~)~o~__
tmiddot bull bull
bull 1
~~ ~
If~il~t -- ~-~ -=I ~of~4j sc J
4 ~ I ol4rlt~ 10 ~
~ -=+1 __ -~ =shy 9~Q 0 0
3 1 bull
flgt ~ o)tl91dmiddot cr~I(~~ 0Ift8l~ fA to1oll it JtfV~J ftshy
-atCf 0) -lOt q~t~t~
o
~ 4O 8 7[ IfJo --shy
pc ~1Qltmiddote1~~ fit It
~1fo1~ Jdt1( t ~) 00 t (- gtj itDI It ~II) 16 it bull bull bull (4 shy
gt( t ( 1middot 10-1 --gt jsmiddot 3 90c ~ -= 3 (lttomiddot ~
~
AAlClt~I~ AvtIlAEvw middot (Awfffvt rt or~t~ 0foifIJV oitlUflff(IIIY)
1f()~~r)JrCl l1ot~ 6u (1~0i1~P~ P1~ 5) fl~ -r~ylrq7~ 11) KAK A-O~~rgt
~ ~ C( ~~ 2~1
f ~ ~----- ~
(~~
l tv UJ 4 ~
4tlo1Q Ill (
~
1 d bull
81 5middot 3
bull bull Af)C1 33j middot~tJ 1~~ A S (jAto
22 ii 0 qDr f Clw Itl) 0 ~-~ 13 5 - 1(bull
G 4 -1shy
~----- 91 ftc
~ 0 -lIshy
t -shy
1 2zl __--shy
bullbull f
10 dfl 0 lt0rotl ~gt TJOfio fiJ1J
~J t J 1lt10 ) 10~ ~ J
10 ~ I
caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
UUrU6tei -= ~ttl 6it Cf ~t ~~) e - f a bull
bull bull amp bull bull -
-shy
I bull
~
a 0
~=
I bull
bull 1 bull I
((1)= 6 IIbull n- 1 r
~ ~~-- - I
C (7 - rmiddotL ~
4 3 J
t o
middotmiddott
r~ 5 1 middot ( 5
- ~ ~ 1 1_ t 0
1T~Q~uIf d~) Il ~1 11 ew ~flgt - rlon) src~ f7f~t(J
61 oflc I t ri1PfW 1tJ)~-1ln~ 6YQl ~ 7 bull )
A V~ J3J middot A c 1~ (Hf1-S It _)7 PI1 (1(f ~ 4 ()-1 I D B (1 f) - 1l f J~o ~~
aOtI4Jfft- Z1 oil bI1 )gt () 8 ~rl A- ((~) 11 ~III7JnlJ 0
1~ - -) rtlffinr(JII~ 0 J J _ C()- 8 fIf IA(A(
b ~
bull It~) ( (1p 11 h ~ I4f iJ 171(Imiddotf~J --C--C-1 1- S) c lt- - - - ---~-~ )tI o~i4 11011 5 t1j1Pf7j~h J~ al 4i~ C ( 9 5) 12 -
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
4 5 - - =- ~6 ampA ~
-8 c
bull Ql1Y~
bull
0
)
rshy
~q
1shy0 () 1-1 1
0 1 0
1 1shy
p ~gtq I
0
~
I
1
o
JA x61= fAl-Il)
IAU~ f lA J t IBf~ IAn B1 - t9t ~fI t va E ~r~IG---___~
A~middotr 1Jrca-10~
hArol~ 10 TI 8~amp~ nxl)
U Un-i~)~o~__
tmiddot bull bull
bull 1
~~ ~
If~il~t -- ~-~ -=I ~of~4j sc J
4 ~ I ol4rlt~ 10 ~
~ -=+1 __ -~ =shy 9~Q 0 0
3 1 bull
flgt ~ o)tl91dmiddot cr~I(~~ 0Ift8l~ fA to1oll it JtfV~J ftshy
-atCf 0) -lOt q~t~t~
o
~ 4O 8 7[ IfJo --shy
pc ~1Qltmiddote1~~ fit It
~1fo1~ Jdt1( t ~) 00 t (- gtj itDI It ~II) 16 it bull bull bull (4 shy
gt( t ( 1middot 10-1 --gt jsmiddot 3 90c ~ -= 3 (lttomiddot ~
~
AAlClt~I~ AvtIlAEvw middot (Awfffvt rt or~t~ 0foifIJV oitlUflff(IIIY)
1f()~~r)JrCl l1ot~ 6u (1~0i1~P~ P1~ 5) fl~ -r~ylrq7~ 11) KAK A-O~~rgt
~ ~ C( ~~ 2~1
f ~ ~----- ~
(~~
l tv UJ 4 ~
4tlo1Q Ill (
~
1 d bull
81 5middot 3
bull bull Af)C1 33j middot~tJ 1~~ A S (jAto
22 ii 0 qDr f Clw Itl) 0 ~-~ 13 5 - 1(bull
G 4 -1shy
~----- 91 ftc
~ 0 -lIshy
t -shy
1 2zl __--shy
bullbull f
10 dfl 0 lt0rotl ~gt TJOfio fiJ1J
~J t J 1lt10 ) 10~ ~ J
10 ~ I
caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
UUrU6tei -= ~ttl 6it Cf ~t ~~) e - f a bull
bull bull amp bull bull -
-shy
I bull
~
a 0
~=
I bull
bull 1 bull I
((1)= 6 IIbull n- 1 r
~ ~~-- - I
C (7 - rmiddotL ~
4 3 J
t o
middotmiddott
r~ 5 1 middot ( 5
- ~ ~ 1 1_ t 0
1T~Q~uIf d~) Il ~1 11 ew ~flgt - rlon) src~ f7f~t(J
61 oflc I t ri1PfW 1tJ)~-1ln~ 6YQl ~ 7 bull )
A V~ J3J middot A c 1~ (Hf1-S It _)7 PI1 (1(f ~ 4 ()-1 I D B (1 f) - 1l f J~o ~~
aOtI4Jfft- Z1 oil bI1 )gt () 8 ~rl A- ((~) 11 ~III7JnlJ 0
1~ - -) rtlffinr(JII~ 0 J J _ C()- 8 fIf IA(A(
b ~
bull It~) ( (1p 11 h ~ I4f iJ 171(Imiddotf~J --C--C-1 1- S) c lt- - - - ---~-~ )tI o~i4 11011 5 t1j1Pf7j~h J~ al 4i~ C ( 9 5) 12 -
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
-8 c
bull Ql1Y~
bull
0
)
rshy
~q
1shy0 () 1-1 1
0 1 0
1 1shy
p ~gtq I
0
~
I
1
o
JA x61= fAl-Il)
IAU~ f lA J t IBf~ IAn B1 - t9t ~fI t va E ~r~IG---___~
A~middotr 1Jrca-10~
hArol~ 10 TI 8~amp~ nxl)
U Un-i~)~o~__
tmiddot bull bull
bull 1
~~ ~
If~il~t -- ~-~ -=I ~of~4j sc J
4 ~ I ol4rlt~ 10 ~
~ -=+1 __ -~ =shy 9~Q 0 0
3 1 bull
flgt ~ o)tl91dmiddot cr~I(~~ 0Ift8l~ fA to1oll it JtfV~J ftshy
-atCf 0) -lOt q~t~t~
o
~ 4O 8 7[ IfJo --shy
pc ~1Qltmiddote1~~ fit It
~1fo1~ Jdt1( t ~) 00 t (- gtj itDI It ~II) 16 it bull bull bull (4 shy
gt( t ( 1middot 10-1 --gt jsmiddot 3 90c ~ -= 3 (lttomiddot ~
~
AAlClt~I~ AvtIlAEvw middot (Awfffvt rt or~t~ 0foifIJV oitlUflff(IIIY)
1f()~~r)JrCl l1ot~ 6u (1~0i1~P~ P1~ 5) fl~ -r~ylrq7~ 11) KAK A-O~~rgt
~ ~ C( ~~ 2~1
f ~ ~----- ~
(~~
l tv UJ 4 ~
4tlo1Q Ill (
~
1 d bull
81 5middot 3
bull bull Af)C1 33j middot~tJ 1~~ A S (jAto
22 ii 0 qDr f Clw Itl) 0 ~-~ 13 5 - 1(bull
G 4 -1shy
~----- 91 ftc
~ 0 -lIshy
t -shy
1 2zl __--shy
bullbull f
10 dfl 0 lt0rotl ~gt TJOfio fiJ1J
~J t J 1lt10 ) 10~ ~ J
10 ~ I
caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
UUrU6tei -= ~ttl 6it Cf ~t ~~) e - f a bull
bull bull amp bull bull -
-shy
I bull
~
a 0
~=
I bull
bull 1 bull I
((1)= 6 IIbull n- 1 r
~ ~~-- - I
C (7 - rmiddotL ~
4 3 J
t o
middotmiddott
r~ 5 1 middot ( 5
- ~ ~ 1 1_ t 0
1T~Q~uIf d~) Il ~1 11 ew ~flgt - rlon) src~ f7f~t(J
61 oflc I t ri1PfW 1tJ)~-1ln~ 6YQl ~ 7 bull )
A V~ J3J middot A c 1~ (Hf1-S It _)7 PI1 (1(f ~ 4 ()-1 I D B (1 f) - 1l f J~o ~~
aOtI4Jfft- Z1 oil bI1 )gt () 8 ~rl A- ((~) 11 ~III7JnlJ 0
1~ - -) rtlffinr(JII~ 0 J J _ C()- 8 fIf IA(A(
b ~
bull It~) ( (1p 11 h ~ I4f iJ 171(Imiddotf~J --C--C-1 1- S) c lt- - - - ---~-~ )tI o~i4 11011 5 t1j1Pf7j~h J~ al 4i~ C ( 9 5) 12 -
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
bull Ql1Y~
bull
0
)
rshy
~q
1shy0 () 1-1 1
0 1 0
1 1shy
p ~gtq I
0
~
I
1
o
JA x61= fAl-Il)
IAU~ f lA J t IBf~ IAn B1 - t9t ~fI t va E ~r~IG---___~
A~middotr 1Jrca-10~
hArol~ 10 TI 8~amp~ nxl)
U Un-i~)~o~__
tmiddot bull bull
bull 1
~~ ~
If~il~t -- ~-~ -=I ~of~4j sc J
4 ~ I ol4rlt~ 10 ~
~ -=+1 __ -~ =shy 9~Q 0 0
3 1 bull
flgt ~ o)tl91dmiddot cr~I(~~ 0Ift8l~ fA to1oll it JtfV~J ftshy
-atCf 0) -lOt q~t~t~
o
~ 4O 8 7[ IfJo --shy
pc ~1Qltmiddote1~~ fit It
~1fo1~ Jdt1( t ~) 00 t (- gtj itDI It ~II) 16 it bull bull bull (4 shy
gt( t ( 1middot 10-1 --gt jsmiddot 3 90c ~ -= 3 (lttomiddot ~
~
AAlClt~I~ AvtIlAEvw middot (Awfffvt rt or~t~ 0foifIJV oitlUflff(IIIY)
1f()~~r)JrCl l1ot~ 6u (1~0i1~P~ P1~ 5) fl~ -r~ylrq7~ 11) KAK A-O~~rgt
~ ~ C( ~~ 2~1
f ~ ~----- ~
(~~
l tv UJ 4 ~
4tlo1Q Ill (
~
1 d bull
81 5middot 3
bull bull Af)C1 33j middot~tJ 1~~ A S (jAto
22 ii 0 qDr f Clw Itl) 0 ~-~ 13 5 - 1(bull
G 4 -1shy
~----- 91 ftc
~ 0 -lIshy
t -shy
1 2zl __--shy
bullbull f
10 dfl 0 lt0rotl ~gt TJOfio fiJ1J
~J t J 1lt10 ) 10~ ~ J
10 ~ I
caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
UUrU6tei -= ~ttl 6it Cf ~t ~~) e - f a bull
bull bull amp bull bull -
-shy
I bull
~
a 0
~=
I bull
bull 1 bull I
((1)= 6 IIbull n- 1 r
~ ~~-- - I
C (7 - rmiddotL ~
4 3 J
t o
middotmiddott
r~ 5 1 middot ( 5
- ~ ~ 1 1_ t 0
1T~Q~uIf d~) Il ~1 11 ew ~flgt - rlon) src~ f7f~t(J
61 oflc I t ri1PfW 1tJ)~-1ln~ 6YQl ~ 7 bull )
A V~ J3J middot A c 1~ (Hf1-S It _)7 PI1 (1(f ~ 4 ()-1 I D B (1 f) - 1l f J~o ~~
aOtI4Jfft- Z1 oil bI1 )gt () 8 ~rl A- ((~) 11 ~III7JnlJ 0
1~ - -) rtlffinr(JII~ 0 J J _ C()- 8 fIf IA(A(
b ~
bull It~) ( (1p 11 h ~ I4f iJ 171(Imiddotf~J --C--C-1 1- S) c lt- - - - ---~-~ )tI o~i4 11011 5 t1j1Pf7j~h J~ al 4i~ C ( 9 5) 12 -
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
0
)
rshy
~q
1shy0 () 1-1 1
0 1 0
1 1shy
p ~gtq I
0
~
I
1
o
JA x61= fAl-Il)
IAU~ f lA J t IBf~ IAn B1 - t9t ~fI t va E ~r~IG---___~
A~middotr 1Jrca-10~
hArol~ 10 TI 8~amp~ nxl)
U Un-i~)~o~__
tmiddot bull bull
bull 1
~~ ~
If~il~t -- ~-~ -=I ~of~4j sc J
4 ~ I ol4rlt~ 10 ~
~ -=+1 __ -~ =shy 9~Q 0 0
3 1 bull
flgt ~ o)tl91dmiddot cr~I(~~ 0Ift8l~ fA to1oll it JtfV~J ftshy
-atCf 0) -lOt q~t~t~
o
~ 4O 8 7[ IfJo --shy
pc ~1Qltmiddote1~~ fit It
~1fo1~ Jdt1( t ~) 00 t (- gtj itDI It ~II) 16 it bull bull bull (4 shy
gt( t ( 1middot 10-1 --gt jsmiddot 3 90c ~ -= 3 (lttomiddot ~
~
AAlClt~I~ AvtIlAEvw middot (Awfffvt rt or~t~ 0foifIJV oitlUflff(IIIY)
1f()~~r)JrCl l1ot~ 6u (1~0i1~P~ P1~ 5) fl~ -r~ylrq7~ 11) KAK A-O~~rgt
~ ~ C( ~~ 2~1
f ~ ~----- ~
(~~
l tv UJ 4 ~
4tlo1Q Ill (
~
1 d bull
81 5middot 3
bull bull Af)C1 33j middot~tJ 1~~ A S (jAto
22 ii 0 qDr f Clw Itl) 0 ~-~ 13 5 - 1(bull
G 4 -1shy
~----- 91 ftc
~ 0 -lIshy
t -shy
1 2zl __--shy
bullbull f
10 dfl 0 lt0rotl ~gt TJOfio fiJ1J
~J t J 1lt10 ) 10~ ~ J
10 ~ I
caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
UUrU6tei -= ~ttl 6it Cf ~t ~~) e - f a bull
bull bull amp bull bull -
-shy
I bull
~
a 0
~=
I bull
bull 1 bull I
((1)= 6 IIbull n- 1 r
~ ~~-- - I
C (7 - rmiddotL ~
4 3 J
t o
middotmiddott
r~ 5 1 middot ( 5
- ~ ~ 1 1_ t 0
1T~Q~uIf d~) Il ~1 11 ew ~flgt - rlon) src~ f7f~t(J
61 oflc I t ri1PfW 1tJ)~-1ln~ 6YQl ~ 7 bull )
A V~ J3J middot A c 1~ (Hf1-S It _)7 PI1 (1(f ~ 4 ()-1 I D B (1 f) - 1l f J~o ~~
aOtI4Jfft- Z1 oil bI1 )gt () 8 ~rl A- ((~) 11 ~III7JnlJ 0
1~ - -) rtlffinr(JII~ 0 J J _ C()- 8 fIf IA(A(
b ~
bull It~) ( (1p 11 h ~ I4f iJ 171(Imiddotf~J --C--C-1 1- S) c lt- - - - ---~-~ )tI o~i4 11011 5 t1j1Pf7j~h J~ al 4i~ C ( 9 5) 12 -
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
rshy
~q
1shy0 () 1-1 1
0 1 0
1 1shy
p ~gtq I
0
~
I
1
o
JA x61= fAl-Il)
IAU~ f lA J t IBf~ IAn B1 - t9t ~fI t va E ~r~IG---___~
A~middotr 1Jrca-10~
hArol~ 10 TI 8~amp~ nxl)
U Un-i~)~o~__
tmiddot bull bull
bull 1
~~ ~
If~il~t -- ~-~ -=I ~of~4j sc J
4 ~ I ol4rlt~ 10 ~
~ -=+1 __ -~ =shy 9~Q 0 0
3 1 bull
flgt ~ o)tl91dmiddot cr~I(~~ 0Ift8l~ fA to1oll it JtfV~J ftshy
-atCf 0) -lOt q~t~t~
o
~ 4O 8 7[ IfJo --shy
pc ~1Qltmiddote1~~ fit It
~1fo1~ Jdt1( t ~) 00 t (- gtj itDI It ~II) 16 it bull bull bull (4 shy
gt( t ( 1middot 10-1 --gt jsmiddot 3 90c ~ -= 3 (lttomiddot ~
~
AAlClt~I~ AvtIlAEvw middot (Awfffvt rt or~t~ 0foifIJV oitlUflff(IIIY)
1f()~~r)JrCl l1ot~ 6u (1~0i1~P~ P1~ 5) fl~ -r~ylrq7~ 11) KAK A-O~~rgt
~ ~ C( ~~ 2~1
f ~ ~----- ~
(~~
l tv UJ 4 ~
4tlo1Q Ill (
~
1 d bull
81 5middot 3
bull bull Af)C1 33j middot~tJ 1~~ A S (jAto
22 ii 0 qDr f Clw Itl) 0 ~-~ 13 5 - 1(bull
G 4 -1shy
~----- 91 ftc
~ 0 -lIshy
t -shy
1 2zl __--shy
bullbull f
10 dfl 0 lt0rotl ~gt TJOfio fiJ1J
~J t J 1lt10 ) 10~ ~ J
10 ~ I
caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
UUrU6tei -= ~ttl 6it Cf ~t ~~) e - f a bull
bull bull amp bull bull -
-shy
I bull
~
a 0
~=
I bull
bull 1 bull I
((1)= 6 IIbull n- 1 r
~ ~~-- - I
C (7 - rmiddotL ~
4 3 J
t o
middotmiddott
r~ 5 1 middot ( 5
- ~ ~ 1 1_ t 0
1T~Q~uIf d~) Il ~1 11 ew ~flgt - rlon) src~ f7f~t(J
61 oflc I t ri1PfW 1tJ)~-1ln~ 6YQl ~ 7 bull )
A V~ J3J middot A c 1~ (Hf1-S It _)7 PI1 (1(f ~ 4 ()-1 I D B (1 f) - 1l f J~o ~~
aOtI4Jfft- Z1 oil bI1 )gt () 8 ~rl A- ((~) 11 ~III7JnlJ 0
1~ - -) rtlffinr(JII~ 0 J J _ C()- 8 fIf IA(A(
b ~
bull It~) ( (1p 11 h ~ I4f iJ 171(Imiddotf~J --C--C-1 1- S) c lt- - - - ---~-~ )tI o~i4 11011 5 t1j1Pf7j~h J~ al 4i~ C ( 9 5) 12 -
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
~q
1shy0 () 1-1 1
0 1 0
1 1shy
p ~gtq I
0
~
I
1
o
JA x61= fAl-Il)
IAU~ f lA J t IBf~ IAn B1 - t9t ~fI t va E ~r~IG---___~
A~middotr 1Jrca-10~
hArol~ 10 TI 8~amp~ nxl)
U Un-i~)~o~__
tmiddot bull bull
bull 1
~~ ~
If~il~t -- ~-~ -=I ~of~4j sc J
4 ~ I ol4rlt~ 10 ~
~ -=+1 __ -~ =shy 9~Q 0 0
3 1 bull
flgt ~ o)tl91dmiddot cr~I(~~ 0Ift8l~ fA to1oll it JtfV~J ftshy
-atCf 0) -lOt q~t~t~
o
~ 4O 8 7[ IfJo --shy
pc ~1Qltmiddote1~~ fit It
~1fo1~ Jdt1( t ~) 00 t (- gtj itDI It ~II) 16 it bull bull bull (4 shy
gt( t ( 1middot 10-1 --gt jsmiddot 3 90c ~ -= 3 (lttomiddot ~
~
AAlClt~I~ AvtIlAEvw middot (Awfffvt rt or~t~ 0foifIJV oitlUflff(IIIY)
1f()~~r)JrCl l1ot~ 6u (1~0i1~P~ P1~ 5) fl~ -r~ylrq7~ 11) KAK A-O~~rgt
~ ~ C( ~~ 2~1
f ~ ~----- ~
(~~
l tv UJ 4 ~
4tlo1Q Ill (
~
1 d bull
81 5middot 3
bull bull Af)C1 33j middot~tJ 1~~ A S (jAto
22 ii 0 qDr f Clw Itl) 0 ~-~ 13 5 - 1(bull
G 4 -1shy
~----- 91 ftc
~ 0 -lIshy
t -shy
1 2zl __--shy
bullbull f
10 dfl 0 lt0rotl ~gt TJOfio fiJ1J
~J t J 1lt10 ) 10~ ~ J
10 ~ I
caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
UUrU6tei -= ~ttl 6it Cf ~t ~~) e - f a bull
bull bull amp bull bull -
-shy
I bull
~
a 0
~=
I bull
bull 1 bull I
((1)= 6 IIbull n- 1 r
~ ~~-- - I
C (7 - rmiddotL ~
4 3 J
t o
middotmiddott
r~ 5 1 middot ( 5
- ~ ~ 1 1_ t 0
1T~Q~uIf d~) Il ~1 11 ew ~flgt - rlon) src~ f7f~t(J
61 oflc I t ri1PfW 1tJ)~-1ln~ 6YQl ~ 7 bull )
A V~ J3J middot A c 1~ (Hf1-S It _)7 PI1 (1(f ~ 4 ()-1 I D B (1 f) - 1l f J~o ~~
aOtI4Jfft- Z1 oil bI1 )gt () 8 ~rl A- ((~) 11 ~III7JnlJ 0
1~ - -) rtlffinr(JII~ 0 J J _ C()- 8 fIf IA(A(
b ~
bull It~) ( (1p 11 h ~ I4f iJ 171(Imiddotf~J --C--C-1 1- S) c lt- - - - ---~-~ )tI o~i4 11011 5 t1j1Pf7j~h J~ al 4i~ C ( 9 5) 12 -
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
1
o
JA x61= fAl-Il)
IAU~ f lA J t IBf~ IAn B1 - t9t ~fI t va E ~r~IG---___~
A~middotr 1Jrca-10~
hArol~ 10 TI 8~amp~ nxl)
U Un-i~)~o~__
tmiddot bull bull
bull 1
~~ ~
If~il~t -- ~-~ -=I ~of~4j sc J
4 ~ I ol4rlt~ 10 ~
~ -=+1 __ -~ =shy 9~Q 0 0
3 1 bull
flgt ~ o)tl91dmiddot cr~I(~~ 0Ift8l~ fA to1oll it JtfV~J ftshy
-atCf 0) -lOt q~t~t~
o
~ 4O 8 7[ IfJo --shy
pc ~1Qltmiddote1~~ fit It
~1fo1~ Jdt1( t ~) 00 t (- gtj itDI It ~II) 16 it bull bull bull (4 shy
gt( t ( 1middot 10-1 --gt jsmiddot 3 90c ~ -= 3 (lttomiddot ~
~
AAlClt~I~ AvtIlAEvw middot (Awfffvt rt or~t~ 0foifIJV oitlUflff(IIIY)
1f()~~r)JrCl l1ot~ 6u (1~0i1~P~ P1~ 5) fl~ -r~ylrq7~ 11) KAK A-O~~rgt
~ ~ C( ~~ 2~1
f ~ ~----- ~
(~~
l tv UJ 4 ~
4tlo1Q Ill (
~
1 d bull
81 5middot 3
bull bull Af)C1 33j middot~tJ 1~~ A S (jAto
22 ii 0 qDr f Clw Itl) 0 ~-~ 13 5 - 1(bull
G 4 -1shy
~----- 91 ftc
~ 0 -lIshy
t -shy
1 2zl __--shy
bullbull f
10 dfl 0 lt0rotl ~gt TJOfio fiJ1J
~J t J 1lt10 ) 10~ ~ J
10 ~ I
caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
UUrU6tei -= ~ttl 6it Cf ~t ~~) e - f a bull
bull bull amp bull bull -
-shy
I bull
~
a 0
~=
I bull
bull 1 bull I
((1)= 6 IIbull n- 1 r
~ ~~-- - I
C (7 - rmiddotL ~
4 3 J
t o
middotmiddott
r~ 5 1 middot ( 5
- ~ ~ 1 1_ t 0
1T~Q~uIf d~) Il ~1 11 ew ~flgt - rlon) src~ f7f~t(J
61 oflc I t ri1PfW 1tJ)~-1ln~ 6YQl ~ 7 bull )
A V~ J3J middot A c 1~ (Hf1-S It _)7 PI1 (1(f ~ 4 ()-1 I D B (1 f) - 1l f J~o ~~
aOtI4Jfft- Z1 oil bI1 )gt () 8 ~rl A- ((~) 11 ~III7JnlJ 0
1~ - -) rtlffinr(JII~ 0 J J _ C()- 8 fIf IA(A(
b ~
bull It~) ( (1p 11 h ~ I4f iJ 171(Imiddotf~J --C--C-1 1- S) c lt- - - - ---~-~ )tI o~i4 11011 5 t1j1Pf7j~h J~ al 4i~ C ( 9 5) 12 -
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
JA x61= fAl-Il)
IAU~ f lA J t IBf~ IAn B1 - t9t ~fI t va E ~r~IG---___~
A~middotr 1Jrca-10~
hArol~ 10 TI 8~amp~ nxl)
U Un-i~)~o~__
tmiddot bull bull
bull 1
~~ ~
If~il~t -- ~-~ -=I ~of~4j sc J
4 ~ I ol4rlt~ 10 ~
~ -=+1 __ -~ =shy 9~Q 0 0
3 1 bull
flgt ~ o)tl91dmiddot cr~I(~~ 0Ift8l~ fA to1oll it JtfV~J ftshy
-atCf 0) -lOt q~t~t~
o
~ 4O 8 7[ IfJo --shy
pc ~1Qltmiddote1~~ fit It
~1fo1~ Jdt1( t ~) 00 t (- gtj itDI It ~II) 16 it bull bull bull (4 shy
gt( t ( 1middot 10-1 --gt jsmiddot 3 90c ~ -= 3 (lttomiddot ~
~
AAlClt~I~ AvtIlAEvw middot (Awfffvt rt or~t~ 0foifIJV oitlUflff(IIIY)
1f()~~r)JrCl l1ot~ 6u (1~0i1~P~ P1~ 5) fl~ -r~ylrq7~ 11) KAK A-O~~rgt
~ ~ C( ~~ 2~1
f ~ ~----- ~
(~~
l tv UJ 4 ~
4tlo1Q Ill (
~
1 d bull
81 5middot 3
bull bull Af)C1 33j middot~tJ 1~~ A S (jAto
22 ii 0 qDr f Clw Itl) 0 ~-~ 13 5 - 1(bull
G 4 -1shy
~----- 91 ftc
~ 0 -lIshy
t -shy
1 2zl __--shy
bullbull f
10 dfl 0 lt0rotl ~gt TJOfio fiJ1J
~J t J 1lt10 ) 10~ ~ J
10 ~ I
caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
UUrU6tei -= ~ttl 6it Cf ~t ~~) e - f a bull
bull bull amp bull bull -
-shy
I bull
~
a 0
~=
I bull
bull 1 bull I
((1)= 6 IIbull n- 1 r
~ ~~-- - I
C (7 - rmiddotL ~
4 3 J
t o
middotmiddott
r~ 5 1 middot ( 5
- ~ ~ 1 1_ t 0
1T~Q~uIf d~) Il ~1 11 ew ~flgt - rlon) src~ f7f~t(J
61 oflc I t ri1PfW 1tJ)~-1ln~ 6YQl ~ 7 bull )
A V~ J3J middot A c 1~ (Hf1-S It _)7 PI1 (1(f ~ 4 ()-1 I D B (1 f) - 1l f J~o ~~
aOtI4Jfft- Z1 oil bI1 )gt () 8 ~rl A- ((~) 11 ~III7JnlJ 0
1~ - -) rtlffinr(JII~ 0 J J _ C()- 8 fIf IA(A(
b ~
bull It~) ( (1p 11 h ~ I4f iJ 171(Imiddotf~J --C--C-1 1- S) c lt- - - - ---~-~ )tI o~i4 11011 5 t1j1Pf7j~h J~ al 4i~ C ( 9 5) 12 -
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
bull 1
~~ ~
If~il~t -- ~-~ -=I ~of~4j sc J
4 ~ I ol4rlt~ 10 ~
~ -=+1 __ -~ =shy 9~Q 0 0
3 1 bull
flgt ~ o)tl91dmiddot cr~I(~~ 0Ift8l~ fA to1oll it JtfV~J ftshy
-atCf 0) -lOt q~t~t~
o
~ 4O 8 7[ IfJo --shy
pc ~1Qltmiddote1~~ fit It
~1fo1~ Jdt1( t ~) 00 t (- gtj itDI It ~II) 16 it bull bull bull (4 shy
gt( t ( 1middot 10-1 --gt jsmiddot 3 90c ~ -= 3 (lttomiddot ~
~
AAlClt~I~ AvtIlAEvw middot (Awfffvt rt or~t~ 0foifIJV oitlUflff(IIIY)
1f()~~r)JrCl l1ot~ 6u (1~0i1~P~ P1~ 5) fl~ -r~ylrq7~ 11) KAK A-O~~rgt
~ ~ C( ~~ 2~1
f ~ ~----- ~
(~~
l tv UJ 4 ~
4tlo1Q Ill (
~
1 d bull
81 5middot 3
bull bull Af)C1 33j middot~tJ 1~~ A S (jAto
22 ii 0 qDr f Clw Itl) 0 ~-~ 13 5 - 1(bull
G 4 -1shy
~----- 91 ftc
~ 0 -lIshy
t -shy
1 2zl __--shy
bullbull f
10 dfl 0 lt0rotl ~gt TJOfio fiJ1J
~J t J 1lt10 ) 10~ ~ J
10 ~ I
caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
UUrU6tei -= ~ttl 6it Cf ~t ~~) e - f a bull
bull bull amp bull bull -
-shy
I bull
~
a 0
~=
I bull
bull 1 bull I
((1)= 6 IIbull n- 1 r
~ ~~-- - I
C (7 - rmiddotL ~
4 3 J
t o
middotmiddott
r~ 5 1 middot ( 5
- ~ ~ 1 1_ t 0
1T~Q~uIf d~) Il ~1 11 ew ~flgt - rlon) src~ f7f~t(J
61 oflc I t ri1PfW 1tJ)~-1ln~ 6YQl ~ 7 bull )
A V~ J3J middot A c 1~ (Hf1-S It _)7 PI1 (1(f ~ 4 ()-1 I D B (1 f) - 1l f J~o ~~
aOtI4Jfft- Z1 oil bI1 )gt () 8 ~rl A- ((~) 11 ~III7JnlJ 0
1~ - -) rtlffinr(JII~ 0 J J _ C()- 8 fIf IA(A(
b ~
bull It~) ( (1p 11 h ~ I4f iJ 171(Imiddotf~J --C--C-1 1- S) c lt- - - - ---~-~ )tI o~i4 11011 5 t1j1Pf7j~h J~ al 4i~ C ( 9 5) 12 -
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
~1fo1~ Jdt1( t ~) 00 t (- gtj itDI It ~II) 16 it bull bull bull (4 shy
gt( t ( 1middot 10-1 --gt jsmiddot 3 90c ~ -= 3 (lttomiddot ~
~
AAlClt~I~ AvtIlAEvw middot (Awfffvt rt or~t~ 0foifIJV oitlUflff(IIIY)
1f()~~r)JrCl l1ot~ 6u (1~0i1~P~ P1~ 5) fl~ -r~ylrq7~ 11) KAK A-O~~rgt
~ ~ C( ~~ 2~1
f ~ ~----- ~
(~~
l tv UJ 4 ~
4tlo1Q Ill (
~
1 d bull
81 5middot 3
bull bull Af)C1 33j middot~tJ 1~~ A S (jAto
22 ii 0 qDr f Clw Itl) 0 ~-~ 13 5 - 1(bull
G 4 -1shy
~----- 91 ftc
~ 0 -lIshy
t -shy
1 2zl __--shy
bullbull f
10 dfl 0 lt0rotl ~gt TJOfio fiJ1J
~J t J 1lt10 ) 10~ ~ J
10 ~ I
caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
UUrU6tei -= ~ttl 6it Cf ~t ~~) e - f a bull
bull bull amp bull bull -
-shy
I bull
~
a 0
~=
I bull
bull 1 bull I
((1)= 6 IIbull n- 1 r
~ ~~-- - I
C (7 - rmiddotL ~
4 3 J
t o
middotmiddott
r~ 5 1 middot ( 5
- ~ ~ 1 1_ t 0
1T~Q~uIf d~) Il ~1 11 ew ~flgt - rlon) src~ f7f~t(J
61 oflc I t ri1PfW 1tJ)~-1ln~ 6YQl ~ 7 bull )
A V~ J3J middot A c 1~ (Hf1-S It _)7 PI1 (1(f ~ 4 ()-1 I D B (1 f) - 1l f J~o ~~
aOtI4Jfft- Z1 oil bI1 )gt () 8 ~rl A- ((~) 11 ~III7JnlJ 0
1~ - -) rtlffinr(JII~ 0 J J _ C()- 8 fIf IA(A(
b ~
bull It~) ( (1p 11 h ~ I4f iJ 171(Imiddotf~J --C--C-1 1- S) c lt- - - - ---~-~ )tI o~i4 11011 5 t1j1Pf7j~h J~ al 4i~ C ( 9 5) 12 -
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
~
AAlClt~I~ AvtIlAEvw middot (Awfffvt rt or~t~ 0foifIJV oitlUflff(IIIY)
1f()~~r)JrCl l1ot~ 6u (1~0i1~P~ P1~ 5) fl~ -r~ylrq7~ 11) KAK A-O~~rgt
~ ~ C( ~~ 2~1
f ~ ~----- ~
(~~
l tv UJ 4 ~
4tlo1Q Ill (
~
1 d bull
81 5middot 3
bull bull Af)C1 33j middot~tJ 1~~ A S (jAto
22 ii 0 qDr f Clw Itl) 0 ~-~ 13 5 - 1(bull
G 4 -1shy
~----- 91 ftc
~ 0 -lIshy
t -shy
1 2zl __--shy
bullbull f
10 dfl 0 lt0rotl ~gt TJOfio fiJ1J
~J t J 1lt10 ) 10~ ~ J
10 ~ I
caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
UUrU6tei -= ~ttl 6it Cf ~t ~~) e - f a bull
bull bull amp bull bull -
-shy
I bull
~
a 0
~=
I bull
bull 1 bull I
((1)= 6 IIbull n- 1 r
~ ~~-- - I
C (7 - rmiddotL ~
4 3 J
t o
middotmiddott
r~ 5 1 middot ( 5
- ~ ~ 1 1_ t 0
1T~Q~uIf d~) Il ~1 11 ew ~flgt - rlon) src~ f7f~t(J
61 oflc I t ri1PfW 1tJ)~-1ln~ 6YQl ~ 7 bull )
A V~ J3J middot A c 1~ (Hf1-S It _)7 PI1 (1(f ~ 4 ()-1 I D B (1 f) - 1l f J~o ~~
aOtI4Jfft- Z1 oil bI1 )gt () 8 ~rl A- ((~) 11 ~III7JnlJ 0
1~ - -) rtlffinr(JII~ 0 J J _ C()- 8 fIf IA(A(
b ~
bull It~) ( (1p 11 h ~ I4f iJ 171(Imiddotf~J --C--C-1 1- S) c lt- - - - ---~-~ )tI o~i4 11011 5 t1j1Pf7j~h J~ al 4i~ C ( 9 5) 12 -
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
1 d bull
81 5middot 3
bull bull Af)C1 33j middot~tJ 1~~ A S (jAto
22 ii 0 qDr f Clw Itl) 0 ~-~ 13 5 - 1(bull
G 4 -1shy
~----- 91 ftc
~ 0 -lIshy
t -shy
1 2zl __--shy
bullbull f
10 dfl 0 lt0rotl ~gt TJOfio fiJ1J
~J t J 1lt10 ) 10~ ~ J
10 ~ I
caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
UUrU6tei -= ~ttl 6it Cf ~t ~~) e - f a bull
bull bull amp bull bull -
-shy
I bull
~
a 0
~=
I bull
bull 1 bull I
((1)= 6 IIbull n- 1 r
~ ~~-- - I
C (7 - rmiddotL ~
4 3 J
t o
middotmiddott
r~ 5 1 middot ( 5
- ~ ~ 1 1_ t 0
1T~Q~uIf d~) Il ~1 11 ew ~flgt - rlon) src~ f7f~t(J
61 oflc I t ri1PfW 1tJ)~-1ln~ 6YQl ~ 7 bull )
A V~ J3J middot A c 1~ (Hf1-S It _)7 PI1 (1(f ~ 4 ()-1 I D B (1 f) - 1l f J~o ~~
aOtI4Jfft- Z1 oil bI1 )gt () 8 ~rl A- ((~) 11 ~III7JnlJ 0
1~ - -) rtlffinr(JII~ 0 J J _ C()- 8 fIf IA(A(
b ~
bull It~) ( (1p 11 h ~ I4f iJ 171(Imiddotf~J --C--C-1 1- S) c lt- - - - ---~-~ )tI o~i4 11011 5 t1j1Pf7j~h J~ al 4i~ C ( 9 5) 12 -
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
caoo 61afPftI tAvtampU1~-B)hv5ul6t D y- Q~ ~f~V bull bull
UUrU6tei -= ~ttl 6it Cf ~t ~~) e - f a bull
bull bull amp bull bull -
-shy
I bull
~
a 0
~=
I bull
bull 1 bull I
((1)= 6 IIbull n- 1 r
~ ~~-- - I
C (7 - rmiddotL ~
4 3 J
t o
middotmiddott
r~ 5 1 middot ( 5
- ~ ~ 1 1_ t 0
1T~Q~uIf d~) Il ~1 11 ew ~flgt - rlon) src~ f7f~t(J
61 oflc I t ri1PfW 1tJ)~-1ln~ 6YQl ~ 7 bull )
A V~ J3J middot A c 1~ (Hf1-S It _)7 PI1 (1(f ~ 4 ()-1 I D B (1 f) - 1l f J~o ~~
aOtI4Jfft- Z1 oil bI1 )gt () 8 ~rl A- ((~) 11 ~III7JnlJ 0
1~ - -) rtlffinr(JII~ 0 J J _ C()- 8 fIf IA(A(
b ~
bull It~) ( (1p 11 h ~ I4f iJ 171(Imiddotf~J --C--C-1 1- S) c lt- - - - ---~-~ )tI o~i4 11011 5 t1j1Pf7j~h J~ al 4i~ C ( 9 5) 12 -
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
r~ 5 1 middot ( 5
- ~ ~ 1 1_ t 0
1T~Q~uIf d~) Il ~1 11 ew ~flgt - rlon) src~ f7f~t(J
61 oflc I t ri1PfW 1tJ)~-1ln~ 6YQl ~ 7 bull )
A V~ J3J middot A c 1~ (Hf1-S It _)7 PI1 (1(f ~ 4 ()-1 I D B (1 f) - 1l f J~o ~~
aOtI4Jfft- Z1 oil bI1 )gt () 8 ~rl A- ((~) 11 ~III7JnlJ 0
1~ - -) rtlffinr(JII~ 0 J J _ C()- 8 fIf IA(A(
b ~
bull It~) ( (1p 11 h ~ I4f iJ 171(Imiddotf~J --C--C-1 1- S) c lt- - - - ---~-~ )tI o~i4 11011 5 t1j1Pf7j~h J~ al 4i~ C ( 9 5) 12 -
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
laquoI bull bullbull
-
I
~ In~ it rli _ I bull )
claquo((l)r)shy (n-1)I - C Cn-j+r r
bull IJ t bull
bull t
bull I bullbull
---shy - ---
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
bull
z1 o~
4 ~r ~ -i to 10 for jW j t~ (
C lf OJ 2) ( 0 1-12-1 ~ Q ~ ~ ~
_ 1JL1-1- S lt 2
t
onof)amp~rJ ~tlU( ~ uno op
bull LJWl ) ~ampIt fJ p ~ 1o~ l5
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
r
amp4Z~-t) I bull shy--shy
bull bull I 1 bull
0 fJ (t PSa (0) ly -to
(r4 traquo)
C ) f
en( ~ ~-r)l r
0
~
t )) tr~ middotB ~ CJ~ (~J r)
JQarro~ t1 ~ ~ fc_
piJL 12 66A0 ~ I
l $J1o
~~ z j P- II~ bull Cot) ~ JJ1JI 6
t oil rtV S 1 UIampt4 AI~ tJMi
(i) ~l~itJo~4t
VOfJi
~I()
-( 1 ~~ t obtl
middotfJtl 1ft lJ I 1 h) POl
- --
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
~) (0 t 1oe)~) ( 411p C (~OJ 5)
(~) AfJtk~ ~llaquo ~ ~ 2 l~ J
)ilvv4 -q
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
19 j12 J 320
~~ I middot ~~I ~ ~~J ~ ~~
(()-tr- 1)l - (r1- )f
p- (q - ) t (err t ) ~ tI ~ t ~ 1) (o ~ bull r~1
ll ~ ~o 6 ~ J4 shy o
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
~JO) - e0tJ - (~- ) (n) ~
(kit) -
o ~ l-lb
k l A~middot~ ~II q n 31) f1 nx ~ bull 1_ 1o() 3
A V Cl~f~rmiddotv4) -Z~l)o Q 3 Q -i -
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
n1 an = 0n- -I- Qn-2 ~ rib QC
lJ 0 Q 1 1
fJill lLf~Cl
lio l (~J )
~ t~ - l eamp(o 020 ~s l 11 IIgt
- PDf (0 7utJ
Jc~UlVo bull
1 )
1 iTf
rt)middot$ bull
~ep 16f-4 TvJ rl f v1A c) eoJ v
ltlot 1 -t ()(z + t ~- i
G bull (0 ~ )+ 01 2~ + 0 t (n1l1
- omiddot qo + ( 1 r ~ t + (-J))
~ middotn ( n- i) 2
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
s
J] ~ i ~ 00 middot
--gt- Z V =7 ~ a -shyW 0 oiO
A
0
(~) -) 1w vJ I JO l
~11 V () C ( )
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
I
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
1 5151 2
cJ (4) 1 ( 5) 2 ri- + 1 z 2 2 f ~
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
middot l~ 1~ ~middot t) W1rT~v 1 ~~t 4 en 7 ~ao1 ( ~~ ~2 ) ~ ArzJ f f (3 Cz))r lt
00
~ 7 f 2 0 01 Z m 11 )
K Al2
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
_p A(z) - _ 1_o fl =i 1 Z
( ~)
O f ~ () +2--Pt ~ Z- 1 1gt 3 lt middot 2 3 A(~) = 3 - -2 ) 2 +- 9 1 - 1- - -J
1 -~2 f - z amp-2)t 1-5 1z
- q 0 - 0 0 1=-CJ 2 i a~ r bull
~ - 1u 1gt =amp0
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
2 zmiddot
J
~ -1 7 11 B- )btv t ~Ol
1v
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
2+1 1fQ~~~ 3 A(z)=z_z+2 I Qn
21 bull_ z 1 A A(2) = z~ut - (tr1)(z-2) -oz- i
z-t1 rA - tgt 1~)(2-Z) ~ L7r +shy -z Joz-1 amp
31 - - A-Ishy ~ G - 1) oz- 2 Z-J
~
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
bull I
- -zbull
~ g(~-L)
~qPf4I~t Av~e li~S rJltt~Us fl 670pound9fPO~gt JJYlf~~l (~ Qrtf i(t-i (tHf -1 + + (0 an ~ n- ~ Ii I f 0 7 7 i- 1
fn ~av~ ~ fa ~oA0 v9 Cl fA v~ ()UJf~~J bull bull
Clil -I ~(rtlJ olCe~9 f ~ = 11 ~ n-1 o~o -) f2 ftf1 -fl ~ 0 Q 11
bull-
n -fJt ~ f~-lt() On ~ (4t L) tfi bull 1
(f 40 -) f ~J U)S c(( seJtICJ ~fmiddotIpoundgt
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
--
QZ -2 ~ - =(Alz) -oo-a-t=J-2i[Mz) -J --He) = D Fgt
Z-2[ A(z) ~ i -22) -2
z-1(
II (z) -1) -gt ~ (io) 0 1=)
A(~)-1- ~z (A(2) 1) _~z2 (f+z)) ~ 0 =-gt AIL) (lt1- tl z - ~ z~) 1 of- 22 - 2z =L
bullbull
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
QttoAo94middotAS fi1 =t 4 bull bull
(11Z ~)
Tof o~tftA ~ ricro oelajAD~ CJnAtJovV Ai- Y~JJ~7IIt__ l 1 p~cS(- (~
[ J 1 0 - ~ ( 1 middotc ~ tl J o middot - ~ ~~
V bull bull middot llaquov )(1 OfJ~+t cP
f fCw on 1c l6ogt tw ---t71 ~t6 7- O J110~rYv ~f9~~~ ~~Ko) rt
Y I 0 tpound) n middot bull
LI ~
J~7 [97 ---) j YJ
t 1 J ~ t1D Q I
1q ~dJ
pound1 [ ACz) -Qo 1 -10 A(z1 = 1 middot 1-~z
A(z~ ~ -4qo ~ ~~ -i(y =
A(z) (1-1OZ) 2 at ~ bull
B+ shy
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
bullbull
1- 1 10
1 ~ -- - 1- _ 1
-~ t
-1 1-~z
aot 1
LIe 0 A (z)= -shy1-10z 3 n
qn = (~o1l) 1011 shy
1
J (z) = bullbull 1-1 oz
lufJ t (J v Av L
A L et ItfQC~ Itti 7 f ~~7rJl1 r Ii 6kyv
~= A ~ =3 ~(2Y-l )al1t1 =-20 l
Xf~61 fflfv11f~V of ZVJV(Aql-A Tfp3P~7~A) o ~)o bull 0 e-- r tgt~OI4 5~otf~q
UiJ 71-] W U ~ n ~ 71 lIo i Y4fr 14 ~
~ C(I () Z (~)
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-
r~o
r
omiddot 0 ~ oj 0
~
lt- Of-dlaquo -
~omiddot~Ct~ middot middot 1s n ifP~~ o (to nf0~ 70 11if)
C ~tl) ~ c (~-41) = t~-4~ (1 +) i () I
- DMath Notes Part1
- DMath Notes Part2pdf
-