1550251515-Classical_Complex_Analysis__Gonzalez_

508 Chapter^7

*19. (a) With the same notation as in Theorem 7.38, show that

( + Z = 1+2 ~ (!:...)n ein(O-.P)

(-z 6 R

(1)

n=l

(b) By introducing (1) into Schwarz's formula (7.28-1) and assuming

that term-by-term integration of the series is valid for r < R (this

will follow from Theorem 4.17 and Corollary 8.1), prove that

00

f(z) = iv(O) +%Ao+ L ( ~) n AneinO (2)

n=l

where An= ~ f

2

7r u(R,'ljJ)e-in.pd'ljJ.

7l" Jo

(c) Letting An= an - ibn, show that (2) decomposes into

1 00 n

u(r,B)= 2ao+ L(~) (ancosnB+bnsinnB) (3)

n=l

00

v(r, B) = v(O) + L ( ~ r (-~n cos nB +an sin nB) ( 4)

n=l

where

an= -^1 127' u(R,'ljJ)cosn'ljJd'ljJ

7l" 0

bn = -^1 127' u(R,'ljJ)sinn'ljJd'ljJ

7l" 0

The Fourier series in (3) and ( 4) represent the harmonic functions

ii(r, B) and v(r, B), respectively, in terms of the simple harmonic

functions rn cos nB = Re zn and rn sin nB = Im zn.

7.29 Application to Fluid Dynamics

In this section we consider an ideal fluid, i.e., a nonviscous fluid in incom-

pressible motion. In other terms, we discuss the motion of a liquid or gas

with no internal friction and with density p independent of the position

vector r of the element of fluid as well as independent of time t.

The velocity field or flow of a fluid is defined by means of a vector

function v(r, t) which gives the velocity of the fluid at every point of a

certain region at every time in some interval. The flow is called steady or

stationary if it is independent of t.