Consider the power series

whose coefficients satisfy the identity

Compute the coefficients and determine the value of .

First, we know the power-series expansion of is given by

So, we have

Equating like powers of we can compute the coefficients and ,

Then to compute the value of we solve the differential equation

If we have

Therefore, . If then we can divide the above differential equation by to obtain the first order linear differential equation

Now, we can solve this differential equation as follows,

Where is an arbitrary constant.

(**Incomplete.** Judging by the answer in the back of the book, Apostol computes this constant as . I’m not sure how to get that though. I think we need some kind of initial condition to determine the constant, and so get a unique solution for . Maybe we can assume this must be continuous at 0 and then take a limit as ? I do think that would get us to , but I don’t know why can assume is continuous at 0. Leave a comment if you have any suggestions.)

How about using the interval -π π, then that f(x) seems to be even. Some jump discontinuities at the endpoints but still integrable.