7.38, §7 More Fun

Our final result was

We can write it directly in terms of the given f(x) if we substitute in the expressions for the Fourier coefficients:

We can clean it up by combining terms and interchanging integration and summation:

This we can clean up even more by giving a name to the function within the curly brackets:

Nice, not? We can even simplify G by converting to complex exponentials and differentiating:

The last because the sums are geometric series.

Integrating and cleaning up produces

So, we finally have the following Poisson-type integral expression giving u directly in terms of the given , with no sums:

Neat!