1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions

Hence if M = max( Mi, M 2 ), we obtain

2n+2M

IRnl < (n + l)! rn+l

555

(8.6-7)

since l~zl < r. But the right-hand side of (8.6-7) tends to zero as n -+

oo, as can easily be seen by considering the convergence of a series with

2n+2rn+1/(n+1)! as general term. Consequently, Rn -+ 0 in Nr(z) as

n -+ oo.

In case 2, by a theorem of J. T. Day [6], we have R 1 ,n -+ 0 and

R2,n -+ 0 as n -+ oo in some Nr1(z) C Nr(z). Hence Rn -+ 0 also in

some neighborhood of z.

For case 3, if f is analytic at z, we have fz = 0, Jlk)(z) = J(k)(z) and

Rn -+ 0 in a circular neighborhood of z, as we have seen in Section 8.3, so

that (8.6-6) reduces to the Cauchy-Taylor series expansion of f(z + ~z).

If f is conjugate analytic at z, then J is analytic and we have

f(z + ~z) = f(z) + f :! ik)(z)~zk

k=l

so that

00 1---

f(z + ~z) = f(z) + L k! J(k)(z)~zk

k=l

in some neighborhood of z.

8.7 Behavior of a Power Series on the Circle of Convergence

We have seen in Section 4.11, Theorem 4.19, that a power series I: anzn
with radius of convergence 0 < R < oo, converges absolutely for every z

such that lzl < R (and uniformly on lzl :::; R 1 < R), and that the series

diverges for lzl > R. Also, we know (Corollary 8.4) that the function

f(z) = I: anzn, lzl < R, is analytic inside the circle of convergence and,

conversely, a single-valued analytic function fat a point a admits a power

series representation f(z) = 2:: an(z - ar valid for lz - al< R, the radius

R being the distance from a to the nearest nonremovable singularity of f

(Theorem 8.5). As to the behavior of I: anzn on the circle lzl = R itself

nothing has been said, and we propose to address this question presently.

Although f(z) = L:anzn, lzl < R, must have at least a singularity son

lzl = R, nothing can be ascertained as to the convergence or divergence

of the power series for z = s, unless some restrictions are imposed on the