30.7 Expressions for Coefficients of a Power Series

We have for the most part so far discussed what to do when confronted with a series. You can test its convergence, estimate its limit, and try to find the function it represents, if it is a power series.

Another important question is:
how can you find the coefficients
in a power series expansion of a given function about some expansion point?

We know from our study of Taylor series in
Section
10.2
that the coefficient of the j-th term will be the j-th derivative of the function at the expansion point, divided by
$j$
factorial.

This is a useful fact, but not always useful enough, in part because it can be cumbersome to calculate or compute the higher derivatives of a complicated function.

Fortunately our standard functions can be defined in the complex plane, and in it we can give an integral representation of the coefficients of a power series, by using the residue theorem.

Suppose we have a function
$f(z)$
and wish to expand it in a series about the point
$z\text{'}$
. We know that the integral of any function around a simple closed path in the complex plane that surrounds an isolated singular point
$z\text{'}$
(and no other singular point) of
$f$
is
$2\pi i$
times its residue at
$z\text{'}$
, and the residue at
$z\text{'}$
is the coefficient of
${z}^{-1}$
in the power series expansion of
$f$
at the point
$z\text{'}$
.

We can therefore deduce that the coefficient
${a}_{n}$
of
${z}^{n}$
in the power series expansion of
$f(z)$
about
$z\text{'}$
, which is the residue of
$\frac{f(z)}{{(z-z\text{'})}^{n+1}}$
at
$z=z\text{'}$
, is
${(2\pi i)}^{-1}$
times the integral of
$\frac{f(z)}{{(z-z\text{'})}^{n+1}}$
on any simple closed path around
$z\text{'}$
that does not include any singular point of
$f$