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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 ' . We know that the integral of any function around a simple closed path in the complex plane that surrounds an isolated singular point z ' (and no other singular point) of f is 2 π i times its residue at z ' , and the residue at z ' is the coefficient of z − 1 in the power series expansion of f at the point z ' .

We can therefore deduce that the coefficient a n of z n in the power series expansion of f ( z ) about z ' , which is the residue of f ( z ) ( z − z ' ) n + 1 at z = z ' , is ( 2 π i ) − 1 times the integral of f ( z ) ( z − z ' ) n + 1 on any simple closed path around z ' that does not include any singular point of f

a n = 1 n ! f ( z ' ) ( n ) = 1 n ! d n f ( z ' ) d z ' n = 1 2 π i ∮ f ( z ) ( z − z ' ) n + 1 d z

Integrals of this kind can be evaluated numerically for any n without great difficulty.

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