1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 591

3. (a) Prove that a convex function e over [xi, X2] is nece~sarily continuous
   in (x1, x2).
   (b) Prove that e" ( x) > 0 is a sufficient condition for e to be convex on
   · [x1,x2].

4. (a) Letting x = 1/ 2 (x 1 + x 2 ) in (8.14-4), we have

e ( X1; X2) < e(x1); e(x2)

Following Jensen [19], this inequality is often taken as the definition of

convexity. However, it is less restrictive than (8.14-4) since it does not

imply continuity.

(b) Letting x = .Ax1 + (1 - .A)x2 in (8.14-4), show that the inequality

becomes

8.15 Series of Negative Integral Powers of z - a

Definition 8.3 Series of the form
00

23 cn(z - a)-n =Co+ c1(z - a)-^1 + · · · + cn(z - a)-n + .' · · (8.15-1)

n=O

where a is a constant and {en} is a sequence of constants, are called series
of negative integral powers of z - a (more precisely, series of powers z - a
with negative integral exponents).

Theorem 8.41 Let r =.limn-+oo V\cJ. Then we have:.

1. If r = O, the series (8.15-1) converges absolutely for every z E C* -{a}.

2. If 0 < r < oo, the series (8.15-1) converges absolutely for every z such

that lz -al > r, the convergence being uniform on lz -al ;:::: r' > r,

and the series is divergent for lz -al < r..

3. If r = oo, the series (8.15-1) is divergent for all finite z.

Proof Letting ( = 1/(z - a), the series (8.15-1) becomes

00

L cnC =Co+ c1( + · · · + cnC + · · · (8.15-2)

n=O

which is an ordinary power series in ( with radius of convergence 1/r. By
Theorem 4.19 we have: