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Page 1: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

WWW.SANJESH.IR 1

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Page 2: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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Page 3: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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Page 4: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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Page 5: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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3 (z g . 4 (z g .

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C C

e ea dz dzi z i z

z : z z z zz

:

z z z zLn ( z) z

z z z

e e e e f ( z )

ze :

z z z z z( ) ( ) ( ) ( ) ( )z zf (z) e! ! ! ! !

z z z z z z zf (z) e

z z z z zf (z) e

3

ee z f (z)

n

nn

zz z z zz( )

z

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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 )( )( )

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Page 28: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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

B

f ( z ) Bz ( z ) z i z i

B

i (B B B ) i ( )

,(z )(z ) i

i i rC : z r r rC .

z z

z i

i

z i

e eB Res f (z)(z )(z ) (z i)(z )

e cos isin(i i)(i ) i ( i )

z z

z i

i

z i

e eB Res f (z)(z )(z ) (z i)(z )

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 )( )

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Page 29: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

z

(z )z

( z ) ( z )

(z ) (z ) (z ) (z ) (z )

(z )

z

(z ) (z )

Bz

ii

z

z

z z ze! ! ! f (z) e sin h

zsin hz !z !z !z

B!z ! ! ! ! ( n)( n )!

z

n

B( n)!( n )!

z ki zf (z)z k

ki (k ) .B .

iaz ia (ki)

iaz ak

z kiz ki

ze (ki)eB Res f (z) e ez ki ki ki

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Page 30: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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 :

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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 .

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Page 32: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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 .

:

y yx xy xy xu cos xe x u cos xe u u

:

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 .

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Page 34: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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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Page 35: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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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Page 36: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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

:

x sa

ts U(x,s) su(x, ) u (x, ) U(x,s) ex

tu(x, ) u (x, ) :

x saU s U e

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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Page 37: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

(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 :

Page 37 of 53

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Page 38: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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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Page 39: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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 .

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Page 40: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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

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Page 41: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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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Page 42: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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

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Page 43: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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

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Page 44: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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

: :

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Page 45: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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 ( ) .

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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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Page 47: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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

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Page 48: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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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Page 49: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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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Page 50: z (1n k nk n kz $$ n (2 n n z $ nn (3 n z $ (4 n z n ai$ n n i b ( n)! b n n a nn ia n ib n Lima i (2 i (1cos( cos ) isin( cos) (4 cos(cos ) isin(cos)(3 ( z)z z ... u u cu xx (D'Alembert)

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

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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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