1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 605

9. Consider the circles lz-al = Ri, lz-bl = R 2 and lz-cl =Ra such that

lb-aJ < Ri +R2, le-bl< R2 +Ra, la-cl< Ri +Ra, determining two

curvilinear triangles ABC and A' B'C' (Fig. 8.11). Let f(z) be analytic

on and inside triangle ABC. Prove that for any z E Int ABC we have

where

f(() d(

(( _ a)n+l'

f(()d(

((-c)-n+l

Also, show that for z E Int A' B' C' the series above converges to zero

(P. Appell [2]).

For another generalization of Laurent series, see D. W. Western

[41]. Other extensions of the Taylor and Laurent series are due to

H. Biirmann and F. Gomes Teixeira. See Chapter 9, Refs. 2 and 7.

8.19 Fourier Series Expansions

Definition 8.5 Let w # 0 be a complex constant, and let S be the strip

bounded by the lines z =a+ wt and z = b +wt, t E ~. Im(b - a)/w < 0

(Fig. 8.12). A function f: S -r <C is said to be periodic in S iff

f(z + w) = f(z)

y

0 x

Fig. 8.12