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Page 1 sur 17

2

2

5

7

8

9

11

12

13

14

15

17

Page 2 sur 17

2

2

1

C

CR

CR

ii²=-1

zC

a+ib ab R

C

C

z=a+ib z

az za Re

bz zb Im

a b c d

00 baiba

db

caidciba

dbicaidciba

bcadibdacidciba

2

21,, eeO

ibaz 2, Rba

baM ,z zM

ibaz baM ,Mz

21 ebeau

z

zu

ibaz u

uz

M OM

ABABzzz

vuvu

zzz

vuzzvu

uu

z.

iyxz xy

iyx zz

1z2z

2121 zzzz

2121 zzzz

2

1

2

12 0

z

z

z

zz

nn zz Zn*Cz

z=a+ib 2, Rba

22 ba zz

zMzOM

ABAzBzAB zzAB

zzz

2

1

2

1

z

z

z

z02 z2121 zzzz

2121 zzzz

nn zz Zn*Cz

zM MO

1e

OM

OMe ,1

OMe ,1

z

zarg

2,arg 1 OMez

1z2z

2argarg 11 zz

2argargarg 2121 zzzz

2argargarg 21

2

1 zzz

z

2argarg znzn Zn*Cz

Page 3 sur 17

2

2

ABCD

AzBzCzDz

2arg,1 AzOAe

AO

2arg,1 AB zzABe

BA

2arg,

AB

AC

zz

zzACABBA CA

2arg,

AB

DC

zz

zzDCABBA CD

3

z

sincos irz 2arg zrz

z

sincos irz ,r

r

b

r

a

bar

ribaz

sin

cos,

22

r,r,

,, rr

,

1

,

1

rr

,,,, ,,, rrrr

,

,,,,

,

,

r

r

r

r

nrr nn,, Zn

n NR

ninin

sincossincos

n,r,rnn

4

ie

ie sincos,1 i

ii ee

i

ie

e

1 iii eee

i

i

i

ee

e inni ee Zn

zirez

2arg zrz

z

2

cos

ii ee

i

ee ii

2sin

5

S02 cbzaz a b c

a acb 42

0

a

b

a

bS

2,

2

0

a

bS

2

0

a

ib

a

ibS

2,

2

1z2z02 cbzaz

a b c a

a

bzz

21

a

czz 21

.

Page 4 sur 17

2

2

6

ABCD

ABCRzz

zz

AC

AB

DCAB //Rzz

zz

AB

CD

0arg

AB

CD

zz

zz

DCAB iRzz

zz

AB

CD

2arg

AB

CD

zz

zz

ABCDRzz

zz

zz

zz

CD

CB

AB

AD

.

ABCA iRzz

zz

AC

AB

ABCAACAB zzzz

ABCA izz

zz

AC

AB

ABC

3;1

AC

AB

zz

zz

ABCDDCAB zzzz

ABCDABCD ADAB

ABCDABCD BDAC

ABCDABCD AB=AC

DCAB zzzz izz

zz

AD

AB

zM '' zM

uu

zzz '

k

zkz'

zez

i'

sincossincos ii

sincossincos ii

sincossincos ii

22sincoscossin ii

xtanxsinxcosx

0010

3321236

122224

323213

102

3232132

1222243

33212365

001

0012

xx coscos

xx sinsin

xx tantan

xx coscos

xx sinsin

xx tantan

xx coscos

xx sinsin

xx tantan

xx coscos 2

xx sinsin 2

xx tantan 2

xx sincos

2

xx cossin

2

xx

tantan

1

2

xx sincos

2

xx cossin

2

xx

tantan

1

2

xxx

xxx

cossinsin

²sin²coscos

22

21122

Page 5 sur 17

2

2

kjiO ,,,

1

a

kzjyixu kzjyixv '''

'''. zzyyxxvu

vuvu 0.

b

kzjyixu

²²² zyxu

c

AB

²²² ABABAB zzyyxxAB

d

P

P

0 dczbyax

cban ,,

P

0 dczbyax

Rt

ctzz

btyy

atxx

e

An

P A n

0. AMnPM

f

PP P

7

a

r

M rM

r rS ,

rMSM

b

cba ;,r

²²²² rczbyax

c

S ABM

0

0

BABABA zzzzyyyyxxxx

BMAMSM .

dE0²²² dczbyaxzyx

4

²

2aa

xaxx

0²²² dczbyaxzyx

222

222

cz

by

ax

dcba

4

²²²

3

0E

0E

2,

2,

2

cba

0E

2,

2,

2

cba

e

rS , r D

H D

Ddd ,

rd

D rS ,

rd H

D

Page 6 sur 17

2

2

rd

D

f

rS , r P

0 dczbyax

H P

222

,cba

dczbyaxPd

rd rS , P

rd rS , PH

P rS ,H

rd rS , P

H22' drr

10d P P rS ,

r

2H

3

a

kzjyixu kzjyixv '''

kjivuyx

yx

zx

zx

zy

zy ''''''

b

0vuuv

c

AB C 0 ACAB

ABC

AB C

ACAB ABC

0. ACABAMABCM

ABC

ABC ACAB

d

ABC 2

ACABS ABC

e

ABCDACABSABCD

d

DA u

u

uAuADd

,;

e

0: dczbyaxP

0'''':' dzcybxaP

cban ,, P

',',' cban 'P

'// PP0' nn

'PP 0'. nn

f

P 'P

n P'n 'P

P 'P'nn

Page 7 sur 17

2

2

1

0n

nn 0nu

0nnnu

nu

0nu 0nnnu

nu 0nnnu

0nnnu

MMun nn 0

0nnnu

m mun nn 0

0nnnu

nunn 0

0nnnu

01 nn uunn 0

0nnnu

01 nn uunn 0

0nnnu

nn uu 1nn 0

0nnnu

r

nruu nn 1nn 0

rnnuu nn 00nn 0

rpnuu pn nn 0pn 0

2

uu1pnu...uu

np

n1pp

n p ;0nnp

0nnnu

q

nnn uqu 1nn 0

0

0.

nn

nn quu

nn 0

pn

pn quu .nn 0pn 0

q

quuuu

pn

pnpp

1

1...

1

1

1qn p ;0nnp

0nnnu

l

l

0nnnu

:a

.

v u wn n n n0

lim limv w l Rn n

un n Ilimu ln

u l vn n n0limvn 0

un n Ilimu ln

u vn nn0limvn

un n Ilimun

v un nn0limvn

un n Ilimun

an

a R

1 1 aliman 0

a 1liman 1

a 1liman

a1 an

nr

r Q *

r 0limn

rn

r 0limn

rn 0

fI : f I Iu0I

u0 u f un n 1n

unl f l l

nn ufv

nulf l

nv lf

Page 8 sur 17

2

2

1

fx0

fx0 00

lim xfxfxx

2

a b,

a b, a b,

ab

R

x xcosx sinxR

x x tan

ZkkRD /2

3

fgx0

f gf g. . fx0

g x0 01

g

f

gx0

4

fIgJ

f I Jx0I

fx0g f x0

g fx0

fx0

fx0

fx0g

J f I J

limx

f x l0

gl

limx

g f x g l0

5

f a b,

f a f bc

a b, f c

f a b, f a f b. 0

f x 0

f a b,

6

fI

IfJ

Iy

xyf

Jx

xfy 1

fI

f1 f If

ff1

7n

n

x xnR

n

x xnR

R

x xnR

x y

y R

n

y x

x R

n

x R x x x R x xn

n nn, , ***

yxyxRyRx nn :

yxyxRyRx nn :

x x x x x

y x y xy

x yx

y

x

yy

n mnm mn nm

n n n

n

n

n

0

0 0

0 0 0

: ****

x :

:

,

,

f I fnI

8

ar

ar

aa ar pq

rp

qp Z *

q N *

a b*

Rr r’ Q*

r

rrrrrrrr

aabaabaaa

1;,. ''

'''

';; rrrrrr

r

r

r

rr

aaaa

a

b

a

b

a

Page 9 sur 17

2

2

1

f a

f a

lf x f x

x aR

x a

lim

lf a f a,

lim ,

x a

f x f x

x af a

2

f a a, 0

fa

limx a

f x f a

x aR

f

a f ad,

f a f ag,

f a

a a

f a f ad g, ,

f a f a f ad g, ,

aa

! x x

f a b,

f a b, a b.a

f a b, a b,b

f a

afaM ; y f a x a f a , .

f a, a

f a

afaM ; y f a x a f a

x a

d

, .

f a

afaM ; y f a x a f a

x a

g

, .

limx a

f x f a

x a

f

a

f a f ad g, , f a

A a f a,

ax

afxf

ax

lim

*Rl 0

fa

f

afaM ;

f

af

afaM ;

f

af

afaM ;

3

f I gJ

JIf a I

f a g af

fg a

afgafafg '''

f I gJ

JIf

f I gJ

fg I x I xfgxfxfg '''

f I

fa 0' af 1f afb

afbf

'

11

Page 10 sur 17

2

2

n1*N. n

*

R 1

' 1

n

n

n

xnxx*

R

n1*N

u I

n uI

1

' '

n

n

n

un

uuxI

4

fI

fI 0' xfxI

fI 0' xfxI

fI 0xI

f0xf’0x

0x

fI

C 0'' xfxI

C

C 0'' xfxI

C

f ‘’0xI0x

00 , xfxIC

C 00 , xfxI

5

fDa b

ax f

x D

xfxaf

Dxa

2

2

ba,f

x D

xfbxaf

Dxa

22

2

TfD

fT

xfTxf

DTxDx :

6

f

c0R

axaR

nx

1*Nn

1nxnR

rx

1 Zn

1rxr*

R *

R

n x

1*Nnn nxn 1.

1

*

R

rx

*Qr

1rxr*

R

x

1

²

1

x

*

R *

R

xx2

1*

R

baxsin baxa cosR

baxcos baxa sinR

xtan

xx

²cos

1²tan1

ZkR

/2

xln

x

1*

R *

R

xexeR

u'uu

vu '' vu uv

vu .'' vuvu uv

v

u

²

''

v

vuvu uv

v

u

1

²

'

u

uu

u

u

u

2

'u

n un nun

u

1.

'

u

nu

1*Nn

'.1 uun nu

ulnu

u'u

ueueu'R

Page 11 sur 17

2

xfax

lim

ax

f

xfxlim y

f

f

الفروع الآلنهائية لمنحنى دالة

a

xfxlim

x

xfx lim

a

f

axy baxy

f

0

axxfx

lim

0lim

baxxfx

baxy f

b

[email protected]

Wadifi

Wadifi

Page 12 sur 17

2

2

1

xx

1

0,

1ln

ln 0,

ln 0,

xy 0,ln ln

ln ln

x y x y

x y x y

xy 0, r Q

( ) ylnxlnxyln +=lny-xlny

xln =

xlnrxln r =-lnyy

ln =1

+∞=+∞→xlnlim

x∞=

→-xlnlim

xx

00

*Nn0=+

nx x

xlnlim

∞→0=

+xlnxlim n

0x→

( )1

1

0=

+

x

xlnlimx→

11

=1-x

xlnlimx→

ln 0,

x xx

01

, , ln,

uI

x u x lnI

x I u xu x

u x, ln

,,

uI

( )

( )xu

xux

,

I

Iλ x u x lnR∈λ

a

a 01 1, , a

0, x xx

aa 0, log

ln

ln

10log

2

ln

exp

expR

xy R ( ) ( ) yxyexpxexp ⇔

( ) ( ) yxyexpxexp == ⇔

x R ( )( ) xxexpln =

x ] [;+∞0( ) xxlnexp =

x R ( ) 0xexp

( ) 10 =exp( ) eexp =1

x R ( ) xexexp =

x y R r Q

yxyx e.ee =+

y

x

y-x

e

ee =

x

x-

ee

1=( )rxrx ee =

( )*Nn∈

+∞=+∞→

x

xelim ; 0=

∞→

x

--xelim

+∞=+∞→

n

x

x x

elim ; 0=

xn

-xexlim

+∞=+∞→ x

elim

x

x; 0=

∞→

x

--xxelim

10

=→ x

1-elim

x

x

xexxexR( ) ( ) x,x ee:Rx =∈∀

uIR( )xuexI

( ) ( )( ) ( ) ( )xu,,xu e.xue:Ix =∈∀

uI

( ) ( )xu, e.xuxII( )

λ+xuexR∈λ

Page 13 sur 17

2

2

f I

f IF

I xfxF ' x I

f IFf I

fI

cxFx c

FG I

cxGxFIxRc :

cx

f I

0xI0y

G f I 00 yxG

G C

I

f g I

FG f g I

GF gf I

FfI

f F fF

0C R

abax

R

nx*Nnc

n

xn

1

1

R

nx

1

1,0Nn

c

xn n

11

1

;0

rx

1,0 Qnc

r

x r

1

1 ;0

x2

1 cx ;0

x

1cx ln ;0

0,

xece x R

axe

ac

a

e ax

R

baxcos a

cbaxa

sin1

R

baxsin

a cbax

a

cos

1R

xx

²tan1²cos

1

cx tan

kk

2;

2

k Z

xuxu r'

1,0 Qr

cxur

r

1

1

1u

xu

xu' cxu 2u

xu

xu

²

'

c

xu

1u

xu

xu ' cxu lnu

xuexu' xueu

Page 14 sur 17

2

2

1

f I F

aFbF F f

ab dxxfb

a aFbFxFdxxf

b

a

b

a

x dxxfb

a

...duufdttfdxxfb

a

b

a

b

a

2

a b c I

dttfdttfdttfb

c

c

a

b

a

0 dttfa

a

dttfb

a dttfa

b3

dttgtfb

a dttf

b

a dttgb

a

dttfdttfb

a

b

a

4

f a dttfxx

a

5

0tft ba,

0 dttfb

a

tgtf t ba,

dttgdttfb

a

b

a

ba

dttfdttfb

a

b

a

ba

abMdttfb

aM

f ba,

6

ba

dttfab

cfbacb

a

1:,

dttfab

b

a

1f ba,

7

u vu’ v’ ba,

dttvtutvtudttvtub

a

b

a

b

a''

8

I

ax bx

jidxxgxfb

a

f

ax bx jidxxfb

a

f

baxy ax bx

audxbaxxfb

a

uajiau

9

.

f ba,

f

uvdttfb

a ²

vu3

iuv

Page 15 sur 17

2

2

1

Ek

1n

2n

kkn

knnn ....21

np*N

pn

pn

pn

pnnp 1

nnn

npNnp 0

En

Eppn

np*N

pnpn

pn

np 1

1....21 pnnnnp

nA

nn

1.2....21 nnn!n1!0

pn!p

A p

n

p

nC

n*Npnp 0

!!

pn

nA p

n

!!

!

pnp

nC p

n

pn

n

p

n CC

1

1

11

P

n

p

n

p

n CCCnp

np

2

n ...,,, 21

A Ap

1)( p

)()()()(),( BpApBApBAPBA 2

),( p

i n;..;21 1

BAAB

),( pAB

0)( p

1)(0 Ap

1)(Ap )(ApAA

BApBpApBAp

pn p

nA p

nC

Page 16 sur 17

2

2

p,

A

card

AcardAp

3

p,AB

0Ap

BA Ap

BAp

BpA ABp

p,AB

0BpAp ABpBpBpBAp

1B2BnB

1B2BnB

nBBB ......21

p,1B2BnB

0,1 1 Bpni

A

n

i

ii BpBApAp1

AB BpApBAp

PAn

k knkk

n ppC

1

4

p,

ix

ba,

p,X

mxxX ,....,1X

ix ixXp

X

p,X

m

k

kk xXpxXE1

.

22 XEXEXV

XVX

5

n 1,0p

X

knkk

n ppCkXpnk

1,0

np

Xnp

npXE

pnpXV 1

Page 17 sur 17

2

2

11

ayy ,

aayy ,

yR

0a0, yy

ayy ,

a

ayy ,axey

R

ayy ,

a0x0a

0

,

xy

ayy

0xxaey

bayy ,

ab

bayy ,

a

bey

ax

R

2

0,,, byayy

ab0,,, byayy

yR

0b0a

0,,, byayy0

, azz

,yz

0b0a0,,, byayy

0,, y xy 2, R

0,,, byayy

ab0b

0,,, byayy

02 barrr

01r2r

xrxreey 21 2

, R

0r

rxexy

2, R

0

iqp iqp

pxeqxqxy sincos 2

, R

02,, yy

xxeey

2, R

02,, yyxxy sincos

2, R

Wadifi