1550251515-Classical_Complex_Analysis__Gonzalez_

502 Chapter 7

the real part of the function on the total boundary of the region. In this
section these results are established for the special case of a circle and for a
half-plane. In Chapter 2 of Selected Topics a more general case is discussed.

Theorem 7.38 Let f be analytic in a region containing the circular disk

lzl ~ R, and let C: ( = Rei.P, 0 ~if~ 271". If the values of u =Ref=
u(R, if) are known, as well as the particular value v(O, B) = v(O), then for

any z == rei^6 with 0 ~ r < R, we have

1 121r ( + z

J(z) =iv(O)+ -

2

u(R,if)-1" -d</>

71" 0 '> -z

(Schwarz) (7.28-1)

Proof By Cauchy's integral formula we have

f(z) = ~ j f(()d( = __!___ 1

2

1r f(()-'-dif

271"Z ( - Z 271" o ( - Z

(7.28-2)

c

since d( = i( dif. In particular, for z = 0 we obtain

f (0) = 2~ 1

2

1r f ( () dif (7.28-3)

For any point z with iz I > R we have

0=~JJ(()d(=__!___1

2

1r f(()-'-dif

27ri ( - z* 271" 0 ( - z*

(7.28-4)

c

If the point z* is chosen to be the inverse of z with respect to the circle C

(Fig. 7.29), i.e., such that zz = R^2 = ((, we get z = ((/z, and (7.28-4)

becomes

0 = -^1 121r f(()_ z - -dif

271" 0 z - '

(7.28-5)

Subtracting (7.28-5) from (7.28-2), we obtain

f (z) = 2~ 121r f ( () [ ( ~ z + ( ~ z] dif

Fig. 7.29