1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

526 CHAPTER 12 • FOURlER SERIES AND THE LAPLACE TRANSFORM

is known as the Poisson kernel. Expanding the left side of Equation ( 12 -15)
in a geometric series gives

1 i(t-9) 00 00
P (r t - 8) =. + re. = " rnein(9-t) + " rnin(t- 9)
' 1 reiCll-t) 1 re•(t-O) L L
n=O n=l
00 00
= 1 + L rn (ein(O- t) + ein(t- 8)] = 1 + 2 L r n cos [n (8 -t) ]
n=l n=l
00
= 1+2 L rn (cosn8cosnt + sinn8sin nt)

00 00

= 1 + 2 L rn cosnBcosnt + 2 L rn sinn8sin nt.

n = l n=l

We now use this result in Equation (12-12) to obtain

u(rcosB,rsin8) = Zn^1 j " _,, P(r,t-8)U(t)dt

1 j" 1 j"

00

=

2

U (t) dt + - L rn cosnBcosnt U (t) dt

7r -1r 7r --w n = l

1 j"

00

+- L:r"sinnBcosnt U(t) dt

1r -11' n=l

= 2 ~ L: U (t) dt + ~: cosnB L: cosnt U (t) dt

• ~r: sinnB L: sinnt U(t) dt

00 00

=a;+ L anrncosnB+ Lbnr"sinn8,

n=l n=l

where {an} and {bn} are the Fourier series coefficients for U (t). This result es-

tablishes the representation for u (r cos8, r sin 8) in Equation (12-11) of Theorem

12.7.

•EXAMPLE 12. 3 Find the function u(x,y) that is harmonic in the unit disk

lzl < 1 and takes on the boundary values

u (cosB,sinB) = U (8) = ~, for -n < B < n.

Solution Using Example 12.1, we write the Fourier series for U (B):

00 c-1r+1

U(t) = L n sinnt.

n=l