Лекции Ивашкин МЖГ

df

=

dz

= lim

z

.

dζ

dζ

ζ→0

ζ

. 36.

" 0" !:

arg lim

dz

= lim arg

dz

= lim arg z − lim arg ζ = arg

df

= α .

(4.26)

dζ

ζ→0 dζ

ζ→0 dζ

w a 2

*

χ* (ζ ) = w∞* ζ +

∞

+

ln ζ .

(4.27)

ζ

2π i

dχ*

=

dχ

dz

=

dχ

f 1 (ζ)

* =

f 1 (ζ ),

(4.28)

w

w

dζ

dz

dζ

dz

"

∞ * =

∞

f 1 (∞) =

∞ m∞ ,

(4.29)

w

w

w

m ∞ – 00 0 .

% 0

w∞

, . .

w∞* w∞ , (4.29) ",

m∞· – !

.

+

"

)*.

! "

(4.6),

. .

* = . . ∫

* dζ = . . ∫

dz

dζ = . . ∫

dz = ,

(4.30)

w

w

w

dζ

c1 *

c1 *

c1 *

( )

( )

w∞ a

2

( )

χ z

= χ * ζ = m∞ w ζ +

+

ln ζ ,

z = f ζ .

(4.31)

ζ

2πi

) " 0"

z = c ch ζ ,

(4.32)

c

– ! ,

x, y

ζ, η.

' , (4.32) ! " " , "

x + iy = c ch (ξ + iη) = c ch ξ cos η + ic

sh ξ sin η;

x = c ch ξ cos η,

y = c sh ξ sin η.

0 " ! !

" 0 "

ξ = α = const, " ! ( . 37)

x 2

+

y 2

= 1

c 2 sh 2 a

c 2 ch 2 a

___________

"

a = c ch α,

b = c sh α

0 "

c = √ a2 – b2.

η = β = const, " !

0 "

"

x 2

y 2

−

= 1

c 2 cos 2 β

c 2

sin 2 β

"

c cos β c sin β.

#, "

0" z = c ch ζ

z-

ζ- ;

) " 0"

z = −

1

iζ 2

2

2

2

x = ξη,

y =

1

(η2 − ξ2 ).

0 " ! !

2

,

ξ = α = const,

" ! η =x / α. %

1 x 2

2

y =

2

− α

α

.

2

,

η = β = const,

" ! ξ = x / β. %

1

2

x

2

y =

β

−

2

2

β

.

P / ργ = const,

. . P = P(ρ).

! :

∂P

=

dP

∂ρ

= a

2

∂ρ

;

∂x

dρ

∂x

∂x

∂P

dP

∂ρ

∂ρ

(5.3)

=

= a 2

,

∂y

dρ

∂y

∂y

∂w

x

+ w

∂w

x

= −

a 2

∂ρ

w

;

(5.4)

x

y

∂x

∂y

ρ

∂x

∂wy

+ wy

∂wy

= −

a 2

∂ρ

wx

.

(5.5)

∂x

∂y

ρ

∂y

(5.4)

wx,

(5.5)

wy

,

:

2 ∂wx

∂wx

∂wy

2

∂wy

a 2

∂ρ

∂ρ

wx

+

+ wy

= −

+ wy

+ wx wy

wx

.

∂x

∂y

∂x

∂y

ρ

∂x

∂y

66

% ,

=

∂ϕ

=

∂ϕ

∂w

x

=

∂

∂ϕ

=

∂ 2

ϕ

w

;

w

;

,

x

y

∂x

∂y

∂y

∂y

∂x ∂x ∂y

(5.6)

(a 2 − w

2 )

∂ 2 ϕ

− 2w

w

∂ 2 ϕ

+ (a 2 − w

2 )

∂ 2 ϕ

= 0.

(5.7)

∂x 2

∂y 2

x

x

y ∂x ∂y

y

)

(5.7)

ϕ = ϕ (x, y)

,

x, y, ϕ,

. &

( ) y = y (x),

(

ϕ

∂ϕ / ∂x

∂ϕ / ∂y. *

.

Au + 2Bs + Ct = 0 ,

(5.8)

A = wx2 – a2; B = wx wy; C = wy2 – a2;

u, s, t

– ϕ.

% u, s, t

wx wy,

. '

P =

∂ϕ

q =

∂ϕ

,

∂x

∂y

Au + 2Bs + Ct = 0 ;

dx u + dy s + 0 t − dP = 0 ;

(5.10)

0 u + dx s + dy t − dq = 0 .

, (5.10)

u, s, t

.

- –

,

–

u,

t =

∂ 2

ϕ

=

t

.

∂y 2

# $ ,

,

u, s, t

.

) , ,

(5.10) 0:

A

2B

C

=

dx dy 0

= → A (dy )

2

− 2B dx dy + C (dx)

2

=

0

dx

dy

(5.11)

dy 2

dy

=

A

− 2B

+ C = 0 .

dx

dx

# ,

= 0

[ . (5.10)]

,

(5.11),

u, s, t

dy

B ± B 2 − AC

= y′

=

.

(5.12)

dx 1,2

1, 2

A

,

= 0.

!!- A, B, C

!!- (5.8),

(5.8) :

δ > 0,

(5.8)

; δ = 0,

;

δ < 0,

– .

),

A = w

2 − a 2

;

B = w

w

y

;

C = w

2 − a 2 ,

(5.14)

x

x

y

δ = a 2 (w 2 − a 2 ).

(5.15)

,

(w > a, δ > 0)

;

(w < a,

δ < 0)

; -

–

(w = a,

δ = 0)

.

. (

dy

1

(w

).

λ

1, 2

=

=

x

w

y

± a w2

− a 2

(5.16)

2 − a 2

dx

wx

(

!

! ,

,

λ

w

( . 38). "

. 38.

α:

# (5.16),

1-e : 1 –

A (x,y),

2 –

x1, y1

!!- λ 1;

2- : 3 –

A

x1 ,

#

A,

4 –

w. $

!!-

λ 2

wx = w,

wy = 0 , :

dy

1

(± a

)= ±

a

1

λ

1, 2

=

= tg α

1, 2

=

w2 − a 2

= ±

.

w2 − a 2

dx

w2 − a 2

Μ 2 − 1

,

α

..

λ

=

dy

= tg (β m α),

(5.17)

1, 2

dx

β – x.

p, q. , (5.13) y΄

A (λ1 q′ − p′)− 2βq′ = 0

(5.18)

1-

p, q.

0

A (λ 2 q′ − p′)− 2βq′ = 0

(5.19)

λ

+ λ

=

2B

.

1

2

a

% 1- (5.18)

Aλ1 = 2B − Aλ 2 ,

A (λ 2 q′ + p′)= 0.

0 2- :

A (λ1 q′ + p′)

(5.20)

(q − q

) = −

1

(p − p

),

0

λ1

0

q0, p0 – ! .

,

y, x

( . 39)

!!-

λ 1,

p, q

–

!!-

–

(1 / λ 1).

,

.

&

p, q,

. 39.

!

.

$ ,

p (x0, y0)