ﺑﺮرﺳﻲ ﺑﺮﺧﻲ از ﺗﻮاﺑﻊ ﻣﺨﺘﻠﻂ e e e =e =e e

ﻂﻠﺘﺨﻣ ﻊﺑاﻮﺗ زا ﻲﺧﺮﺑ ﻲﺳرﺮﺑ

1 - ﻲﺋﺎﻤﻧ ﻊﺑﺎﺗ f(z) e= z

( )

x+iy x

x x

f(z) = e e cos(y) + i sin(y) u e cos(y) , v e sin(y)

=

= =

( )

( )

z+2πi x+iy+2 i x

x z

e = e e cos(y + 2 ) + i sin(y + 2 ) = e cos(y) + i sin(y) = e

π = π π

z x z

| e | e , arg(e ) y= =

x x

x x

u -e sin(y) u e cos(y)

x

v= e co y

v = e sin(y) s( ) y

x y

∂ =

∂

∂

∂

∂ =

∂

∂

∂

ﻊﺑﺎﺗ ﻦﻳا ﺲﭘ مﺎﺗ

ﺖﺳا .

x x

z

u v

w = + i w e cos(y) + i e sin(y)

x x

= e = w

∂ ∂

′ → ′=

∂ ∂

تﺎﺒﺛا :

( )

u u

u u

w u+iv u

u u

u u

U u v

e cos(v) + e (-sin(v))

y y y

V u v

= e sin(v) + e (cos(v)) x

U u v

e cos(v) + e (-sin(v))

x x x

e = e = e cos(v) + i s

V u v

e sin(v) + e (co in(v)

s(

x x

= U +

) i

y y

V

y v )

∂ = ∂ ∂

∂ ∂ ∂

∂ ∂ ∂

∂

∂ ∂ ∂

∂ = ∂ ∂

∂ ∂ ∂

∂ = ∂

∂ ∂

∂

ﺲﭘ :

U V

, U y

y x

x

- V

∂ ∂

∂

∂

∂ = ∂

∂ = ∂

ﻲﺋﺎﻤﻧ ﻊﺑﺎﺗ صاﻮﺧ زا ﻲﺧﺮﺑ :

1

1 2 1 2 1 2

2

z

z z z +z z - z

z

z c cz w(z) w(z)

e e = e e = e e

(e ) = e [e ] = e

′

w (z)

′

ﺖﺴﻴﻧ ﺮﻔﺻ ﻪﻄﻘﻧ ياراد ﻊﺑﺎﺗ ﻦﻳا .

2 - ﻲﺗﺎﺜﻠﺜﻣ ﻊﺑاﻮﺗ

iz -iz iz -iz

e - e e + e

sin(z) = , cos(z) =

2i 2

sin(z) cos(z)

tg(z) = , cotg(z) =

cos(z) sin(z)

و سﻮﻨﻴﺳ ﻊﺑﺎﺗ اﺬﻟ ،هدﻮﺑ ﻲﻠﻴﻠﺤﺗ ﺎﺟ ﻪﻤﻫ ﻲﺋﺎﻤﻧ ﻊﺑﺎﺗ ﺪﻨﺘﺴﻫ ﻲﻠﻴﻠﺤﺗ ﺰﻴﻧ سﻮﻨﻴﺴﻛ .

[ ] [ ]

d sin(z) cos(z) d

cos(z) - s

dz in(z)

= dz =

sin(z) = sin(x + iy) = sin(x)cos(iy) + cos(x)sin(iy)

ﻂﺑاور زا 1 و 3 ﻢﻳراد :

تﺎﺒﺛا : sin(z) = sin(x)cosh(y) + i cos(x)sinh(y) = 0

( ) ( )

ix - y -ix + y

-y y

e - e z = x + iy sin(z) =

2i

e cos(x) + i sin(x) - e cos(x) - i sin(x) sin( )

2i

→

⎡ ⎤

⎣ ⎦

= z

2 2 2

2 2 2

2 2 2 2

| sin(z) | = sin (x) + sinh (y)

| cos(z) | = cos (x) + sinh (y) sin (z) + cos (z) = cosh (y) - sinh (y) = 1

⎧⎪

⎨⎪

⎩

sin(x)cosh(y) = 0 sin(x) = 0 x = n cos(x)sinh(y) = 0 sinh(y) = 0 y 0

⎧ ⇒⎧ ⇒⎧

⎨ ⎨ ⎨ =

⎩ ⎩ ⎩

π cos(z) = cos(x)cosh(y) - i sin(x)sinh(y) = 0

(

^{tg(z) =}

)

¹₂

cos (z) ′

3 - ﻢﺘﻳرﺎﮕﻟ ﻊﺑﺎﺗ f(z) Log(z)=

z = rei^θ → f(z) = Log(r) + iθ

2 2 -1

2 2 -1

1 y

f(z) Log(x + y ) + i tan ( )

2 x

1 y

u Log(x + y ) v tan ( )

2 x

=

= =

2 2 2

2 2

2 2

2 2

2

u y

y x + y

v -y

x x + y

u x

x + y 0 x x x + y

v x

y x + y

, y 0 z 0

∂ = ∂ =

∂

≠ ⇒ ≠

∂ =

∂

∂ =

→

∂

≠ ∂

2 2 2 2 2 2 2

x y x - iy z z

f (z) = - i = = =

x + y x + y x + y | z | zz f (z) = 1

z

′

′

ﻢﺘﻳرﺎﮕﻟ ﻊﺑﺎﺗ صاﻮﺧ زا ﻲﺧﺮﺑ :

2

1 2 1 2

1

1 2

2 z

1 2 1

Log(z z ) = Log(z ) + Log(z ) Log( ) = Log(z ) - Log(z )z

z

Log(z ) = z Log(z )

لﺎﺜﻣ 1

i : w = i

πi 2

-π

i 2

Log(w) i Log(i) iLog(e ) i ( i) -

2 2

w = i = e

= = = π = π

لﺎﺜﻣ 2 :

i

z = -1 = e1 ^π ⇒Log(-1) = iπ

4 - نوراو ﻲﺗﺎﺜﻠﺜﻣ ﻊﺑاﻮﺗ

-1

iw -iw

iw -iw

2iw iw iw 2

2 2

1 2

w = sin (z) sin(w) = z e - e

z = e - e - 2iz = 0

2i

e - 2ize -1 = 0 e = iz ± 1- z

w = -i Log(iz + 1- z ) , w = -i Log(iz - 1- z )

→

⇒

⇒

داد نﺎﺸﻧ ناﻮﺗ ﻲﻣ

1 2 :

w + w = (2k +1)π

-1

2 -1

1

2

2

- 1

sin ( 1

cos (z) = -1 z)

tg (z

1 =

1-

) 1+

z

= z

- z

′

′

⎡ ⎤

⎣ ⎦

⎡ ′

⎡ ⎤

⎣

⎣ ⎤⎦

⎦

5 - ﻲﻟﻮﻟﺬﻫ ﻊﺑاﻮﺗ

z -z z -z

e - e e + e

sinh(z) = , cosh(z) =

2 2

sinh(z) cosh(z)

tgh(z) = , cotgh(z) =

cosh(z) sinh(z)

cos(x)cosh(y) = 0 cos(x) = 0 x = n -sin(x)sinh(y) = 0 sinh(y) = 0 y 0 2

⎧ +

⎧ ⇒⎧ ⇒⎪

⎨ ⎨ ⎨

⎩ ⎩ ⎪ =⎩

π π

(

^{cotg(z) =}

)

^-1₂

sin (z)

′

1 2 1 2 1 2

Log(z z ) = Log(z ) + Log(z ) , z = z = -1

[ ] [ ]

d sinh(z) = cosh(z) d

cosh(z) = sin dz

dz h(z)

( ) ( )

(x+iy) -(x+iy)

x -x

e - e

sinh(z) = 2

1 e cos(y) + i sin(y) - e cos(y) - i sin(y)

2⎡ ⎤

= ⎣ ⎦

ﻲﻟﻮﻟﺬﻫ ﻊﺑاﻮﺗ يﺎﻫﺮﻔﺻ ﻦﻴﻴﻌﺗ

sinh(z) = sinh(x)cos(y) + i cosh(x)sin(y) = 0 sinh(x)cos(y) = 0 sinh(x) = 0 x = 0 cosh(x)sin(y) = 0 sin(y) = 0 y n

⎧ ⎧ ⎧

⇒ ⇒

⎨ ⎨ ⎨ =

⎩ ⎩ ⎩ π

cosh(z) = cosh(x)cos(y) + i sinh(x)sin(y) = 0

x = 0 cosh(x)cos(y) = 0 cos(y) = 0

sinh(x)sin(y) = 0 sinh(x) = 0 y n 2

⎧ ⇒⎧ ⇒⎧⎪

⎨ ⎨ ⎨ = +

⎩ ⎩ ⎪⎩ π π

6 - نوراو ﻲﻟﻮﻟﺬﻫ ﻊﺑاﻮﺗ