I’m studying this question from Kröner: given $f:\mathbb{R}^2\rightarrow\mathbb{R}$,
$$f(x,y)=\sum_{n=1}^\infty\frac{(-1)^n}{n^2}\sin nx\sin ny.$$
I need to calculate $f(\frac{3\pi}{4},-\frac{5\pi}{4})$. I know I can brutally solve it by injecting the value into the series, but I’m trying to extract the function $f(x,y)$. I tried transforming the equation into something like $\cos n(x-y)-\cos n(x+y)$, and seperate $f$ into 2 functions of parameters $(x-y)$ and $(x+y)$, but I’m stuck here.
Indeed $2f(x,y)=g(x-y)-g(x+y)$, where $$g(x)=\sum_{n=1}^\infty(-1)^n\frac{\cos nx}{n^2}=s(x+\pi),\qquad s(x)=\sum_{n=1}^\infty\frac{\cos nx}{n^2}.$$ The last series is well-known (see e.g. this post). Another idea is that $s(x)$ and $g(x)$ are integrated sawtooth waves; there's a more general connection with Bernoulli polynomials, also seen here.
As a result, for $|x|\leqslant\pi$ we get $g(x)=\frac{x^2}{4}-\frac{\pi^2}{12}$, thus $f(\frac{3\pi}{4},-\frac{5\pi}{4})=-\frac{\pi^2}{32}$.