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Theorems about Power Series

Taylor series

The coefficients of the power series

for the function f(z) are given by:

Example:

where

Convergence

If a function has a power series expansion around some point a, then the circle of convergence
extends to the nearest point at which the function is not analytic. (Analyticity is a technical
term which you will learn about in PH 461. Brie y, a function which is not analytic is
singular in some way. A function is certainly not analytic at any point at which its value
becomes infinite or at a branch point of a root.)

Uniqueness

The power series of a function, if it exits, is unique, i.e. there is at most one power series of
the form which converges to a given function within a circle of convergence
centered at a. We call this a power series "expanded around a".

Note: This theorem is an open invitation to collect a bag of cute tricks. It doesn't matter
how you find a series for a function, once you have it, it is the series. The rest of these
theorems should be in your bag of cute tricks.

1. A power series may be differentiated or integrated term by term. The resulting series con-
verges to the derivative or integral of the function represented by the original series within
the same circle of convergence as the original series.

Example:

2. One series may be substituted in another provided that the values of the substituted series
are in the circle of convergence of the other series.

Example:

What happens if you try this same trick to find a power series for 1/(1 + cos z)? Why?

Another example:

Note: This is a very short power series with just two non -zero terms.

Note: Starting with a power series for sin(z) expanded around z = 0, we have obtained a
power series for cos(z) expanded around .

3. Two power series of like powers may be added, subtracted , or multiplied . The resulting
series converges at least within the common circle of convergence.

Example:

Compare this to the result you would get using the previous theorem. Which method is
faster
?

Another example:

Compare this series to the series for the function(see the first example in
Theorem 2.) What can you conclude about the wisdom of as suming two series are the same
if their first three terms are identical?

4. Two power series expanded around the same point may be divided . If the leading term(s) of
the denominator series is not zero, or if the zero(s) is canceled by the numerator , then the
resulting series converges within some circle. If the radius of convergence of the numerator
and denominator series are and , respectively, and the distance from the origin of the
circles to the nearest zero of the denominator series is s, then the quotient series converges
at least inside the smallest of the three circles of radii , and s.

Try the previous example sin z/(1 + z) using synthetic division , instead. Is this method
easier or harder? Imagine what you would do if the denominator were a power series with
an infinite number of non-zero terms.

5. The series expansions for most functions recorded in books are expansions around the point
z = 0. To expand around a point a ≠ 0 write every z which appears in the function as
(z - a) + a, simplify creatively , and use Theorem 2.

Example: Expand sin z around z =π .

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