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TRANSCRIPT
WWW.SANJESH.IR 1
Page 1 of 53
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zzzArg( )z
1 ((z ) 2 ( Re zRe z
3 (Im z
Im z
4 (g
f (z) f (x iy) u(x, y) iv(x, y)Du
y
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2 (u(x, y) x .
3 (u(x, y) ax b a b .
4 ( u(x, y) .
vu
1 (yv yu . 2 (xv xu .
3 (yu yv . 4 (xu xv .
uG
x yA z G | u (z) u (z)u
1 (u G .
2 ( u u .
3 ( u A u .
4 ( u A u .
Page 2 of 53
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D (x, y) | x , yze
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4 (ire : e r e ,
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2 (f (z) D .
3 (f (z) D .
4 (f (z) D .
Page 3 of 53
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fz | Im z
1 (f (z) a sin z bcos z a b .
2 (f .
3 (f (z)P(z)
P(z)
.
4 (f .
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Page 4 of 53
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( )z
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Page 5 of 53
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z
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Page 6 of 53
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x x Lf (x)
L x L x L
n nn
a n x n xf (x) a cos b sinL L
:
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n
sin x xf (x)
xb
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sin x xf (x)
cos x xb
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Page 7 of 53
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(D'Alembert)u xxu c u
xtu (x, ) exu(x, ) eu ( , t) u(L, t)
1 (xu(x, t) e cos hct sin hctc
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c
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u u sin xu( , t) , u ( , t)u(x, ) f (x)
w
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3 ((x) cos x x 4 ((x) sin x x
xy xu u
1 (y(y sin x)e x 2 (x(x sin y)e y
3 (x(y sin x)e x 4 (y(x sin x)e y
Page 8 of 53
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x
,tx t
u( , t) , u (L, t)u(x, ) f (x)
f (x)
1 (x x ( k ) xcos , cos , , cos ,L L L
2 (x x n xsin , sin , , sin ,L L L
3 (x x ( k ) xcos , cos , , cos ,L L L
4 (x x ( k ) xsin , sin , , sin ,L L L
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t t
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1 (n tnu(x, t) E e sin(nx)
2 (n t
nnu(x, t) E e sin
3 (n t
nnu(x, t) E e cos x
4 (n tn
Au(x, t) E e cos(nx)
Page 9 of 53
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t xx
x
u u , x L , tu(x, ) (x) , x Lu(L, t) , u ( , t) , t
(x)
1 (K xcos , K NL
2 (x x x ( k ) xcos , cos , cos , , cos ,L L L L
3 (x x x ( k ) xsin , sin , sin , , sin ,L L L L
4 (x x x n xcos , cos , cos , , cos ,L L L L
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2 ( r a A(r) nr cos sin .
3 ( r a A(r) nr ln r .
4 ( r a A(r) nr ln r .
L mT (x, ) sin( x)
T Tt x
T( , t)T( , t)
T( / , )
1 ( 2 (/ 3 (/ 4 (/
xx yyu ur au(a , ) sin
1 (a cosr
2 (r cosa
3 (r sina
4 (a sinr
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u u , x yx y
u(x, y) x y , x y
u(x, y)
1 (max minu , u 2 (max minu , u
3 (max minu , u 4 (max minu , u
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u (r, )
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au(a, ) f ( )u
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Page 11 of 53
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g(t)tG(s)
t xxu u g(t) t ,u (x , ) , u( , t)
U(x ,s)U(x ,s)
1 (x sG(s)( e )
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3 (x sG(s)( e )s
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1 (cos (a w)a w
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4 (sin (a w)a w
Page 12 of 53
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f (t)t
i tf ( ) f (t)e dt
f (t)
1 (f ( ) e 2 (f ( ) e
3 (f ( )i
4 (f ( )i
cos x dxx
1 (xe 2( xe 3 (xe 4 (xe
f (x)cos x dx
1 (f ( x ) sin xd 2 (f ( x ) cos xd
3 (f ( x ) sin xd 4 (f ( x ) cos xd
xf (x) ex
mx sin mx dxx
1 (me 2 (me 3 (me 4 (me
xe
1 ( 2 ( 3 ( 4 (
xf (x) ex
1 (cf 2 (cf
3 (cf 4 (cf
Page 13 of 53
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fxsin x x
f (x)x
i xf̂ ( ) f (x)e dx
1 (sin 2 (i sin 3 (sin 4 (i sin
w sin wxI dww
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n
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1 (z i 2 (z i 3 (z i 4 (z i
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n
n
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1 (2 2 ( 3 ( 4 (4
w f (z)zf (z)
1 ( i .
2 ( .
3 ( .
4 ( z .
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3 (f (z) D .
4 (f (z) D .
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1 (xy x xy 2 (xy x xy
3 (xy x xy 4 (xy x xy
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1 (b , a 2 (b ,a
3 (a b 4 (a b
Page 15 of 53
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P Ax Bxy Cy Dyz Ez Fxz
1 (E (A B) 2 (E (A C)
3 (E A B 4 (E A c
f (r, ) r cos r sin
1 ( .
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3 ( .
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f (z) ww
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3 ( v V .
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z ni z nin
1 ( 2 ( 3 ( 4 (
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zz
1 (Re(z z ) 2 (Im (z z ) z z
3 (Re (z z ) z z 4 (k
zz
1 ( z
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h
f (z nih) f (z)Limh
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f (x iy) x xyi
1 ( 2 (
3 ( 4 (
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Page 17 of 53
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w (z )z
z
1 (
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3 ( .
4 ( .
zzz
1 ( 3 .
2 ( 3 .
3 ( 3 .
4 ( 3 .
log z i
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Page 18 of 53
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3 ( h D .
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C
e dzz ( z )
1 (i 2 (i 3 ( 4 (i
z
Ce dzC
1 (ie ( e) 2 (ie ( e) e
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f (z)sin hz
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f (z) Ln( z)z
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1 (1 2 (3 3 (4 4 (5
Page 19 of 53
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z
dzz a
aa
1 ( 2 (a
3 (1 4 (a
x dxx
1 ( 2 ( 3 ( 4 (
Page 20 of 53
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iz z e cos isin
i
i
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( cos ) sin isin (sin )i i sinA( cos ) sin cos ( cos ) cos
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X
sinY( cos ) cos
sin
sinTan Arg(A)( cos ) sin
sin Im(z) 4 .
x y
y x
u vf
u v - f f (z) u iv
u y y xu v u f (x) cv f (y) c
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x
y
u f (x) cv f (y) c
v cy c u cx c
Page 21 of 53
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y x xy yy y xx yy
y y yy xy x xx yy
U u U u v V U UV v U u v V V V
V U .
1 ( 2 ( 4 ( 3 .
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y
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u .
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x y
y x
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xr e x e r e
y y
i i z zz e z e ,z z z
i ie (e ) (cos i sin )
(cos ) isin (cos ) i sin(cos ) i sin (cos ) sin cos cos sin
(cos ) isin i sin( cos ) ( cos )
i sin(cos i sin ) ( cos )
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i sin isin u iv( cos ) ( cos )
u .
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w .
f 1 3 f f
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u v f
f (z) m m M f (z) u v m
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y y x x x
uu vv uu vv uu(u v )u
uu vv v uv vu v
v c u c f (z) u iv . f (z) f (z) C
z z | Im(z) . f (z) C .
Page 23 of 53
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u . xx yyu u . G x
y
uu
u .
x yu u(x, y) f (y) u f (y) f (y) C
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G .
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g(z)
h .
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f (z) Cg(z).
f (z) :
f (z)z z
z z . f (z)z( ) z( )
z z
:z
z
Page 24 of 53
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z :
nn
nn n
nn
n n
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z
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n
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n
kkn
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n
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n n n
k kn
k n n
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nn nLim a Limi i i i
(cos ) Ln i i coscos( ) (cos )Logi
nnLima i e e e
nnLima cos ( cos ) i sin ( cos )
Page 25 of 53
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zf (z) ( z)
z
C C
f (z) ( z)a dz dzi z i z
z
z z zzLn ( z) zz
C C
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:
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z z z z zf (z) e
3
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n
nn
zz z z zz( )
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Page 26 of 53
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n n n
n nn n nz z z z z
iz e
z :
n
n ni in
n ne
e e (cos ) isin
n
n
(cos ) i sin (cos n isin n )(cos ) isin cos ) isin
n n
n n
(cos ) i sin cos n i sin n(cos ) sin (cos ) sin
2 :
n
n
nn
n n
(cos ) cos ncos
sinsin sin sin ncos
cos
n
nn
sin sin sincoscos
n z g .
(z ) (z )
ii
( 2 )( )( )
Page 27 of 53
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z :
A Bz C D Ef (z)z (z ) z i z i
A B C D E z z f (z)(z ) (z )
:
A , B , C , D , E
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i
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e cos isin( i i)(i ) i (i )
cos i sin cos isinI i (B B ) ii(i ) i(i )
(i ) cos i(i )sin (i ) cos i(i ) sinI ii(i )
i cos isin iI (cos sin )( )
Page 28 of 53
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z
(z )z
( z ) ( z )
(z ) (z ) (z ) (z ) (z )
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Bz
ii
z
z
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n
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ki (k ) .B .
iaz ia (ki)
iaz ak
z kiz ki
ze (ki)eB Res f (z) e ez ki ki ki
Page 29 of 53
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iaz
iaz ak akzef (z)e dz dz i e iez k
:
akx sin ax dx ex k
f . (z ) (z )
i
A Bz C D Ef (z)z (z ) z i z i
z
f (z)z (z ) z i z i
A B C D E :
f :
z (z )
(z ) (z ) z (z )
f (z)z (z ) z i z i
B , B , B
C
f (z) dz i (B B B ) i
f (x) nb T L :
Page 30 of 53
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nna f (x)cos x dx cos nx dx sin nx sin n
n n
n
a f (x)dx dx f (x) sin n cosnxn
x :
n n
sin nf ( ) sin n cos( )n n
n
sin nn
:
nb
L
nna x cos x dx
L L
L
nxL n L n L La sin x cos x cos n
L n L n L L n n
n (even)L (cos n ) Ln n (odd)
n
f (x) nb :
T L L L
nna f (x)cos x dx
L L
a f (x) cos x dx cos x dx ( )dx
sin x sin sin
f (x) x x 1 .
Page 31 of 53
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L
n L
nb f ( x )sin x dx b f ( x )sin xdxL L
cos xsin x sin x dx ( )sin x dx dx
sin xx ( )
b f (x)sin x dx sin x sin x dx cos x sin x dx
x sin x cos x
u(x, t) f (x ct) f (x ct) G(x ct) G(x ct)c
x xG(x) g(x)dx e dx e
x ct x ct x ct x ctu(x, t) e e e ec
x
x xee ch(ct) sh(ct) e ch(ct) sh(ct)c c
x t
x t
t xu(x, t) f (x t) f (t x) g( )dx
L
L
L Lu , f ( L) f ( L) g( )f
L
L
L L L Lu , g( )d
g( ) ( L)( L ) L L
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L
L
L L Lu , L L
L L L L L L L
L LL L
L L
t xx xxw w sin x
xx xx xsin x sin x cos x A
sin x Ax B
4 .
:
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:
x x xx xy xy xu xe x sin y e u cos ye u u
:
x x xx xy xy xu cos x e y sin x e x u ye u u
:
y yx xy xy xu x cos x e u x cos x e u u
: :
xu v :
y y yy xv v v A(x)e u A(x)e u B(x)e C(y)
4 .
Page 33 of 53
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u( , t) x
xu (L,T) x L
.
:g(x) f (x) x c :
x ct
x ctu(x, t) f (x ct) f (x ct) g( )d
c
(x t) (x t) x t
2 1
u( , t) 3 .
x L
2 .
1 .
T( , t) T( , t) n tn
nb e sin n x
T(x , ) sin x :
n tn
nsin x b e sin n x
sin x b sin x b sin x b sin x
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b nb . :
tT(x , t) e sin x T( / , ) e sin /e
u(a , ) sin 1 2 .
r r a
4 3 .
u(x, y) .
u x y x y .
x y x y .maxu
y x y x .minu
3 .
:
nn n
nu(r, ) (A cos n B sin n ) r
u u(a , ) :
a f ( )d d
r u .
sU(x,s) u(x, ) U(x,s) G(s)x
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U sU G(s)x
sx sxhs s s U (x,s) A(s)e B(s)e
s ,x
lim U(x,s) B(s) :
sxhU (x,s) A(s)e
U sU G(s)x
PG(s)U
s :
sx G(s)U A(s)es
:
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ts U(x,s) su(x, ) u (x, ) U(x,s) ex
tu(x, ) u (x, ) :
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x
s s sxe sxe
. :
x x xs s sa a a
PU e e eD s s ss aa
x ssx sx aU(x,s) c e c e es
a
jwxF(w) f (x)e dx
(a w )ix
a ix iwx eF(w) e e dxa i iw
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(a w )i (a w )iF(w) e e(a w)i
sin( a w)F( w) sinin( a w)( a w)i a w
.
f (x) g( )
:
f (x) g( )cos x dx cos x d
1 . .
xcf f (x)cos x dx e cos x dx
:
cs
sL cos x fs
2 .
2 4 :f̂ ( ) f (x)(cos x isin x)dx
f f (x) cos x dx :
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f̂ ( ) f (x) ( isin x)dx f (x)( i sin x)dx
i sin x sin x dx
:
cos x sin x sin xsin xsin xsin wxdx sin x cos x dx
sin sinsin x sin x dx sin x sin x dx
sin isinf̂ ( ) i
:
w sin wxI dww
ww
:
ixz
Im z
w sin wx zeA dw Im dzw z
f (z) z i z i :
ixz ixz x
z i z i
ze ze eResz z
ixz x
x
Imz
ze edz i iez
:xeI .
izz ez
:
z z i
z i .
Page 38 of 53
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n .
z :
i i iz e dz ie d dz ie d i(cos i sin ) d
sin i cos d ( sin ) (cos ) d d
:
z
I z dz d
z x iy , z x iy , dz dx idy
dx dy x y :
:
n
nnn
(z i) z ilim z i
k
n kn(kn)! ~e
:
n
nn nn n n n
( n)!( n)!lim a(n,z) lim z lim z(n!) (n!)
n
nelim z z zne
R .
Page 39 of 53
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n n )
( .
v u .
u v v xy xx y y
u v v x y yx x x
y :
v xy xy f (x)
:
y y f (x) x y y f (x) x f (x) x c
v xy xy x
x xx
y yy
u ax by u axu bx u
u(x, y) :
xx yyu u ax a
:
p p px y z
Page 40 of 53
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p pAx By Fz , Ax x
p pBx Cy Dz , Cy y
p pDy Ez Fx , Ez z
:
A C E E (A C)
:
i iz re r e r (cos i sin )
f (r, ) r (cos i sin ) r cos z .
( )( ) ( )( )
z :
( )( ) ( ) ( )( ) ( ) ( )( ) ( )
.
*
(*)
:A :
A ( )( )
A
(*) :
A
A A
Page 41 of 53
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zf (z) e w
iw re :
k
ln w r i ( k )z
z kr i( k )
k .
z ni z ni z ni z ni
(z ni)(z ni) (z ni)(z ni)
zz niz niz n zz niz niz n niz niz z z
) ( x x iy x iy y
z x iy A x y
x y z y x y z x :
xA x z x
xA x .
xx z x xA
z x z x
z x x z x x x z
zxzz x x x
zy
Page 42 of 53
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A . A
. .
xA xA .
x
x x( ) z x ( z x x)z x z xA
z x
x xA x(x ) z x (x )A
z xz x ( z x )
maxz zA z x
zA (x )z
xA
.
i
i
z rez z z z z z ( z z )
z r e
(z z )(z z ) z z z z
z z z z z z z z z z z z z z z z z z
: z z z z z z Re(z z ) z z
i( )Re(z z ) z z Re(r r e ) r r
r rr r cos( ) r r k
x iyz zz x iyz z x iy
(x ) iyz (x ) yz (x ) y(x ) iy
(x ) y (x ) y
x x y x y x
Page 43 of 53
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x y x x x y
(x ) ( ) y (x ) y ( )
(x ) y ( )
: ( , ).
z
zf (z) Lim(z e )
.
z z ze z! !
z z z z zz e z z! ! ! !
zLimf (z)
x y x .
.
: f (z) z .
f (z) z x iy
f (z) x yf (z)
( ) f (z) ( )(x y )
x y x y
A f (z) A A
:A
: :
Page 44 of 53
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f (z) u iv
f (z) u v f (z) f (z) (u v ) (u v )x y x y
x xf (z) u iv
- A.
.
. .
h h
f x i(y nh) f (x yi)f (x iy nih) f (x yi)Lim Limh h
yh
f (x, y nh) f (x, y)Lim nfh
: y h
f (x, y h) f (x, y)f Limh
: h
f (x, y nh) f x, y (n )h f x, y (n )h f (x, y)Lim
h
y h
f x, y (n )h f (x, y)f Lim
h
nfy .
z
(x,y) ( , )
xy(x iy) f (z) f ( )f (x iy) f ( ) limx y z
xy(x iy)lim(x iy)(x y )
y x
(x,y) ( , ) x
xy x( x)f ( ) lim f ( ) limx y x x
f ( ) f ( ) .
Page 45 of 53
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z .
z .
z z
z z z zlim lim( z z z z )z
( ) ( ) ( ) ( )
.
.
. f g z a g(a)
:
z a z a z a
f (z) f (z) f (z)lim lim limg(z) g (z) g (z)
z z
z z z z z z zlim limz
z z z z z
z z z z z z
z zz z
z zz z
z zz z
zz
iz z e x iy
w i
Page 46 of 53
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i
i i ie
w (z ) w ( e ) e ez
(cos i sin ) (cos isin ) w cos i sin
xcosx yr w i
y r rsinr
x y r :
x x rr ( ) ( )y y rr
y x .
3 .
z z z zAz z z z
j j j
z x iyz x iy j , ,
L : ax by c
z, z ,z ,z L .
j i
j i
ax by c a(x x ) b(y y )ax by c j , , j , ,
j j
i j i j
ay y (x x )b
ay y (x x )b
i j i, j , ,.
: j j j
j j j j
z z (x iy) (x iy )
a(x x ) i(y y ) (x x ) i (x x )b
Page 47 of 53
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j j
j j
i j i ji j i j
az z (x x ) iz z x xbz z x xaz z (x x ) i
b
.
iz re log z ln z i arg (z) iln r i( k ) i
ln r r r
k k ( k)
i i ( k ) ( k )iz re e e
iz iz log i iz (ln i i arg i) ii e e , (z re x iy)
iz ( i ( k )) ( k )z ( k )(x iy)izi e e e
( k )( x iy)iz
( k )(x iy) ( k )zi e k
e e
( k )ye ize :
sin x x k (k )x k
x z .
x iyu ivx iy x y
xux y
yvx y
z x iy (x ) iy (x ) y
Page 48 of 53
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z i x iy i x i(y ) x (y )
x y xx y y
xy
Xx y x ux y
yx y y vx y
h(z) , z
h ( z ) h(z) h .
h h(z) C C C . h
C h(C) C C.
zh(z) z D z z
h .zh(z) h( )
f C : z zef (z)
z
(n)
nC
f (z) if (z)dz(z z ) n! z
z .
(*)
z
( )
C
eiz dz f ( ) if ( )
z !
z z z ze e (z ) ze e (z )f (z) f (z)
z (z ) (z )
Page 49 of 53
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z z ze (z ) (z )e (z ) (z )( z)e (z )
f (z)(z )
i (*)f ( )
z x
Ce dz e dx (e )
C z xz x iy , x , z x
dz dx
z iy iy i
Ce dz ie dy e e
C
z iyz x iy y
dz idy
z x i i x i i i
Ce dz e dx e (e ) e ( e) e e
Cz x iy z x i , x , dz dx
z iy iy i
Ce dz ie dy e e
Cz x iy z iy , y , dz idy
z z z z z
C C C C C
i i i i z i
C
e dz e dz e dz e dz e dz
(e ) ( e ) (e e ) ( e ) e dz e ( e) e
azf (z) e C : z :
x ixz z e
z C f :
az
C C
f (z) edz i f (z ) dz i f ( )z z z
az
a
C
ef (z ) e dz iz
Page 50 of 53
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ixz e :
ixdz ie dx izdx
z cos x i sin x
.
iz a (cos x isin x)
C
e edz i (iz dx) iz z
a cos x ia sin xe e dx
iasin xe cos(a sin x) isin(a sin x)
a cos xe cos( a sin x ) i sin( a sin x ) dx
:
a cos x a cosxe cos(a sin x)dx i e sin(a sin x)dx i
a cosxe cos (a sin x)dx
zn n
n n
.
z :z z . z .
log n( z ) ( z )z n e
n
.
n
n
z z zsin hz(n )! ! !
z zz z zsin hz ! ( !) !! ! !
Page 51 of 53
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.
z z z! ! !
z z z z z z z! ! ! ! ! ( !) !
z z z z! ! ! !
z z z! ( !) ! !
z z z z z! ( !) ! ! ! !z z z z! ! ! ( !) ! !
z z z z z( !) ! ! ! ! ( !)
z z :z z z z zz
:
C
dz z zzz
z , f (z) z , g(z) z z
f (z) z
f (z) g(z)g(z) z z
f f g ) ( z . 5
.
z :
g(z)f (z)
z f (z) g(z)g(z) z z z f (z)
Page 52 of 53
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f f g ) ( z .f .
z .
z a f (z)z a
a .
z a z az a
d (z a)Res f (z) lim lim( )( )! dz z a
z zf (z)z
ki( )
kz e k , , , , ,
i
e i
e i
e .
i i
z e z e
zB Res f (z)z
i
z e
iB z e i
i
z e
B z e
i
i i
z e
iB z e e
B B cos isin cos isin
B B B B B
i (B B B ) i ( )
Page 53 of 53
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