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.)