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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
2²
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
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