Home  18.013A  Chapter 30 


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 jth term will be the jth 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 2i times its residue at z', and the residue at z' is the coefficient of in the power series expansion of f at the point z'.
We can therefore deduce that the coefficient in the power series expansion of f(z) about z', which is the residue of at z = z', is times the integral of on any simple closed path around z' that does not include any singular point of f
Integrals of this kind can be evaluated numerically for any n without great difficulty.
