nmbr complex

7/29/2019 Nmbr Complex

1/17

Page 1 sur 17

2

25789111213141517


2/17

Page 2 sur 17

2

21

CCR C

i=-1za+ib

CC

z=a+iba za Reb zb Im

abc00 baiba

db

caidciba

dbicaidciba

bcadibdacidciba 2

21,, eeO

ibaz 2, Rba baM , zM

ibaz baM ,M

z

21 ebeau

zu

ibaz u

uz

MOMABAB

zzz

vuvu

zzz

vuzzvu

uu

z.

iyxz xy

iyx z

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 z

zzOM AAzBzAB zzAB

zzz

2

1

2

1

z

z

z

z02 z2121 zzzz

2121 zzzz nn

zz Zn *Cz

z MO 1e

OM

OMe ,1 OMe ,1z

zarg

2,arg 1 OMez

1z2z

2argarg 11 zz 2argargarg 2121 zzzz

2argargarg 212

1 zzzz

2argarg znzn Zn *Cz


3/17

Page 3 sur 17

2

2A

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

nN nini n sincossincos

n,r,r nn4

i

e

i

e 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 cbzazabcaacb 42

0

a

b

a

bS

2,

2

0

a

b

S 2

0

a

ib

a

ibS

2,

2

1z2z02 cbzaz

abca

bzz

21

a

czz 21


4/17


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Page 5 sur 17

2

2 kjiO ,,,

1a

kzjyixu kzjyixv '''

'''. zzyyxxvu

vuvu 0.

bkzjyixu

zyxu c

A ABABAB zzyyxxAB d

PP

0dczbyax cban ,,

0dczbyax

Rtctzz

btyy

atxx

eAn

Pn0. AMnPM

f

7a

MrM

rS ,rMSM

b cba ;,

rczbyax c

S AB

00

BABABA zzzzyyyyxxxx

BMAMSM .

d0 dczbyaxzyx

4

2

2aa

xaxx

0 dczbyaxzyx

222

222

cz

by

ax

dcba

4

300

2

,

2

,

2

cba

0

2,

2,

2

cba

e rS , D

H D Ddd ,

rd D rS ,

rd D


6/17

Page 6 sur 17

2

2 rd

D

f rS , r P

0 dczbyax

H P

222,

cba

dczbyaxPd

rd rS , Prd rS , PH

P rS ,Hrd rS , P

H22' drr 10d P P rS ,

r

2

3a

kzjyixu kzjyixv ''' kjivu yx yx

zxzx

zyzy ''''''

b0 vuuv cAC0ACAB

ABCAC

ACAB ABC 0. ACABAMABCM

ABC

ABC ACAB d

ABC2

ACABSABC

eABCDACABSABCD

d Du

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


7/17

Page 7 sur 17

2

21

0n

nn 0nu

0nnnu nu

0nu

0nnnu

nu 0nnnu

0nnn

u Mun nn 0

0nnn

u mun nn 0 0nnnu

nunn 0

0nnnu 01 nn uunn 0

0nnn

u 01 nn uunn 0

0nnn

u nn uu 1nn 0

0nnn

u

nruu nn 1nn 0 rnnuu nn 00 nn 0 rpnuu pn nn 0pn 0 2

uu1pnu...uu

np

n1pp

np;0nnp

0nnn

u

nnn uqu 1nn 0 0

0.

nn

nn quunn 0 pn

pn quu .nn 0pn 0

q

quuuu

pn

pnpp

11

...

1

1

1qp;0nnp

0nnn

u l

0nnn

u :a

.

v u wn n n n0

lim limv w l Rn n un n Ilimu ln

u l vn nn0limvn 0 un n Ilimu ln

u vn nn0limvn un

n I

limun v un nn0limvn un n Ilimun

ana R

1 1 aliman 0a1liman 1a 1lima

n a 1an

nrr Q*

r 0limn

rn r 0lim

n

rn 0

fI:f I Iu0Iu0u f un n1n

unlf l l nn ufv nu nv lf


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Page 8 sur 17

2

21

fx0

fx0 00

lim xfxfxx

2 a b,

a b,a b,ab

R

x xcosx sinxR

x xtan

ZkkRD /2

3fgx0

f gf g. . fx0

g x0 01g

f

g

x0

4fIgJ

f I Jx0Ifx0gf x0

g fx0

fx0

fx0

fx0g

Jf I J lim

xf x l

0

gl lim

xg f x g l

0

5fa b,

f af bc a b,f c

fa b, f a f b. 0 f x 0

fa b,6

fI

IfJ

Iy

xyf

Jx

xfy 1fIf1 f If

ff17

nx x

nRn

x xnRRx 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

y

y

n mnm mn nm

n n n

nn

n

0

0 0

0 0 0

: ****

x :

:

,

,

fIfnI8

ar

araa ar pq

rp

qp Z*q N*

*RrQ*

r

rrrrrrrr

a

abaabaaa1

;,. ''

''''

;; rrrrrr

r

r

r

rr

aaaa

a

b

a

b

a


9/17

Page 9 sur 17

2

21

ff l f x f xx a Rx a

lim

l f a,

lim ,x a

f x f x

x af a

2f a a, 0

f limx a

f x f a

x aR

fa f ad,

fa f ag,

f

a f a f ad g

, ,

fa f a f ad g, , a

!x xf a b,

f a b, a b.af a b, a b,

f afaM ; y f a x a f a , .

f a, af

afaM ; y f a x a f ax a

d

,.

f afaM ; y f a x a f a

x a

g

,.

lim

x a

f x f a

x a

a f a f ad g

, ,fa

A a f a,

ax

afxf

ax

lim

*Rl 0f

f

afaM ;

fa

afaM ;

fa

afaM ;

3

f JIf I

f affg

afgafafg '''

f JIf

ffgI xfgxfxfg '''

ff 0' af

1f afb afbf '11


10/17

Page 10 sur 17

2

2n1*N

*R 1'

1 nnn

xnx*R

n1*Nu

n u

1

' '

n

n

n

un

uu

4ff 0' xff 0' xf

f0xf0x0x

0x

fC 0'' xfCC 0'' xfC

f 0x0x 00 , xfxIC 00 , xfxI

5fb

ax

xD

xfxaf

Dxa

2

2

ba,xD

xfbxaf

Dxa

22

2

Tf

xfTxf

DTxDx :

6f

c

axnx

1*Nn1nxn

rx

1 Zn1rxr*R *R

n

x1*Nnn nxn 1.

1

*

R

rx

*Qr

1rxr*R

x

1

1

x

*R *R

xx2

1*

R

bax sin baxa cos bax cos baxa sin

xtanx

xcos

1tan1

ZkR

/2

xlnx

1*R *R

xe

xe

u'u

vu '' vu vu .'' vuvu

v

u

''

v

vuvu v

u

1

'

u

u

u

u

u

2

'

n un nun

u

1.

'

u

nu

1*Nn'.

1 uun nu

ulnu

u '

ue

ueu'


11/17

Page 11 sur 17

2

xfaxlim ax

f

xfxlim y

f

f

a

xfxlim x

xf

x

lim

a

f

axy baxy

f

0

axxfx

lim

0lim baxxfxbaxy f

b

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12/17

Page 12 sur 17

2

21

x x

1

0,1ln

ln 0,

ln 0,

xy 0,ln ln

ln ln

x y x y

x y x y

xy 0,rylnxlnxyln +=lny-xln

y

xln =

xlnrxlnr=-lny

yln =

1

+=+ xlnlimx= -xlnlim

xx

0

0

*Nn 0=+

nx

x

xlnlim

0=+

xlnxlim n

0x

( ) 110

=+

x

xlnlimx

11

=1-x

xlnlimx

ln 0, x x

x0

1, , ln

,

u x u x ln

x I u x u xu x, ln,

,

u

( )

( )xu

xux

,

I

I x u x lnR

aa 0 1 1, ,

0, x xx

aa 0, log

ln

ln

10log

2ln exp

expxR( ) yxyexpxexp

( ) yxyexpxexp ==

xR( )( ) xxexpln =x;+0( ) xxlnexp =

xR( ) 0xexp( ) 10 =exp( ) eexp =1

x( ) xexexp =xRr

yxyx e.ee =+

y

x

y-x

e

ee =

x

x-

ee

1=( )rxrx ee =

( )*Nn

+=+

x

xelim= x--x elim

+=+ n

x

x x

elim 0= xn-x exlim

+=+ x

elim

x

x;= x--x xelim

10

= x

1-elim

x

x

xex

xex ( ) ( ) x,x ee:Rx =

uIR( )xu

ex I

( ) ( )( ) ( ) ( )xu,,xu e.xue:Ix =uI

( ) ( )xu, e.xux II( )

+xu

ex R


13/17

Page 13 sur 17

2

2f

fI xfxF '

ff cxFx

F cxGxFIxRc :c

f0x0y

GI 00 yxG GC

I

fgFgGFgf Ff

f 0a

bax n

x*Nn c

n

x n

1

1

Rn

x

1 1,0Nn

c

xn n

11

1

;0

rx

1,0 Qncrx r

1

1 ;0

x2

1cx ;0

x

1cx ln ;0

0,xecex ax

e

ac

a

e axR

bax cosa

cbaxa

sin1R bax sin

a cbax

a

cos

1Rx

xtan1

cos

1

cx tan

kk2

;2

k

xuxu r' 1,0 Qr

cxur

r

1

1

1

xuxu' cxu 2

xuxu

'

c

xu

1

xuxu ' cxu ln

xuexu' xue


14/17

Page 14 sur 17

2

21

f aFbF

adxxfba aFbFxFdxxf bab

a

x dxxfba ...duufdttfdxxf b

a

b

a

b

a 2

abc dttfdttfdttf bc

c

a

b

a 0 dttfaa

dttfba

dttfab

3

dttgtfba dttfba dttgba dttfdttfba

b

a4

fdttfx xa

5 0tft ba, 0 dttfba tgtf t ba,

dttgdttf ba

b

a ba

dttfdttf ba

b

a ba

abMdttfba

f ba,

6ba

dttfab

cfbacb

a

1:,

dttfab

b

a

1f ba,

7uvv ba,

dttvtutvtudttvtu bababa '' 8

Iax bx

jidxxgxfba

f

ax bx jidxxfb

a f baxy ax bx

audxbaxxfba

uajiau

9.

f ba,f

uvdttfba

vu3

iuv


15/17

Page 15 sur 17

2

21

Ek

1n

2n

kkn

knnn ....21

n*Np

pp

pnp 1n

nNnp 0E

E

n*Nppn

pnp 1

1....21 pnnnnpnAn

1.2....21 nnn!n1!0

p!p

Apn

p

nC

n*Nnp 0 !

!

pn

nApn

!!

!

pnp

nCpn

pnn

pn CC

1111 Pnpnpn CCCnp

n2

n ...,,, 21

A Ap1)( p

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

),( p

i n;..;21 1 BAAB

),( pAB

0)( p

1)(0 Ap

1)(Ap )(ApAA BApBpApBAp

pn

p

nA p

nC


16/17

Page 16 sur 17

2

2 p,

A

card

AcardAp

3

p, 0ApB

ApBAp

BpA ABp

p, 0BpAp ABpBpBpBAp

1B2BnB

1B2BnB

nBBB ......21

p,1B2BnB

0,1 1 Bpni

A

n

i

ii BpBApAp1

AB BpApBAp P

k knkkn ppC 1

4

p,

ix

ba,

p, mxxX ,....,1

ix ixXp X

p,

m

k

kk xXpxXE1

.

22 XEXEXV

XVX

5

n 1,0pX

knkk

n ppCkXpnk

1,0 n

X npXE

pnpXV 1


17/17

Page 17 sur 17

21

ayy ,

aayy ,yR

0a0, yy

ayy ,a

ayy ,axey

R

ayy ,a0x0a

0

,

xy

ayy

0xxaey bayy ,

ab

bayy ,a

bey ax

R

20

,,, byayyab0

,,, byayyyR

0b0a

0,,, byayy0, azz

,yz0b0a0,,, byayy

0,, yxy

2, R

0,,, byayy

ab0b0

,,, byayy0

2 barr

01r2r

xrxr eey 21 2, R0r

rxexy 2, R

0

iqpiqp pxeqxqxy sincos 2, R

02,, yyxx eey

2, R02,, yyxxy sincos 2, R

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nmbr complex

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