1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 597

Substitution of (8.18-6) and (8.18-10) into (8.18-3) gives

00 00

f(z) = L An(z-at+ L A_n(z - a)-n

n=O n=l

(8.18-12)

m=-oo

where the coefficients An (n = 0, 1, 2, ... ) are given by formula (8.18-7) and

the coefficients A_n (n = 1, 2, ... ) by formula (8.18-11).
Since the function f(t) is analytic in r' ~ it -al ~ R', as well as the
integrands f(t)/(t -at+\ f(t)/(t -a)-n+i, the contours C1 and C 2 may

both be continuously deformed into a contour r contained in the ring and

such that the winding number of r with respect to a be +1. In particular,

we may take for r the circlet-a= r 0 ei^9 (0 ~ (} ~ 27r), with r' ~ r 0 ~ R'.

Hence both formulas (8.18-7) and (8.18-11) may be combined into

Am= _1 J f(t)dt

27l'i (t -a)m+l

(8.18-13)

r

where m = 0, ±1, ±2, ....

Theorem 8.43 The series in (8.18-12) converges absolutely in r < lz-al <

R, and uniformly on the ring r^1 ~ iz -al~ R', where r^1 and R' are any

two positive real numbers such that r < r' < R' < R.

Proof By definition

+oo oo oo

. L IAm(z -arl = L IA-n(z-a)-nl + L IAn(Z -atl
m=-oo n=l n=O

Since the series z::::=l A-n(z - a)-n converges absolutely for Jz - al > r

and the series z::::=o An(z - a)n converges absolutely for lz -al < R, so
does the series L:~:_ 00 Am(z - ar for any z such that r < iz - al< R.
To show uniform convergence on any smaller closed ring, let
n 00
Sn= l:A 11 (z-at, fi(z) = L Av(z -aY
v=O v=O
and
n' 00
Tn' = LA-v(z-a)-^11 , h(z) = L A-v(z - a)-^11
v=l v=l