Principles of Mathematics in Operations Research

180 13 Power Series and Special Functions

Theorem 13.3.2 Suppose the series J2^=ocnxn converges for \x\ < R, and

define

oo

f(X) = YlcnX

n

, \X\<R

n=0

which converges uniformly on [—R+e,R — e], no matter which e > 0 is chosen.

The function f is continuous and differentiable in (—R,R), and

oo

f'{x) = Yjncn{x-a)n-\ \x\<R

n=l

Corollary 13.3.3 / has derivatives of all orders in (—R,R), which are given

by

f(k)(x) = J2 n{n - 1) • • • (n - k + l)cn(x - a)n~k.

n=k

In particular,

f(k\0) = k\ck, k = 0,1,2,...

Remark 13.3.4 The above formula is very interesting. On one hand, it shows

how we can determine the coefficients of the power series representation of f.

On the other hand, if the coefficients are given, the values of derivatives of f

at the center of the interval (—R,R) can be read off immediately.

A function f may have derivatives of all order, but the power series need

not to converge to f(x) for any x ^ 0. In this case, f cannot be expressed as

a power series about the origin.

Theorem 13.3.5 (Taylor's) Suppose, f(x) = Y^=ocnXn, the series con-

verging in \x\ < R. If —R < a < R, then f can be expanded in a power series

about the point x = a which converges in \x — a\ < R— \a\, and

/(„-i;ffl2>(,-.,-.

n=0

Remark 13.3.6 If two power series converge to the same function in (-R, R),
then the two series must be identical.

13.4 Exponential and Logarithmic Functions

We can define
oo n

n=0
It is one of the exercise questions to show that this series is convergent Vz G C.
If we have an absolutely convergent (if |uo| + |ui| + • • • is convergent) series,
we can multiply the series element by element. We can safely do it for E(z):