I have come across the following in my textbook and I am a little confused by it
$$\cosh(\pi y )\cos(\pi x) = \sum_{n=0}^\infty q_n(y) \cos(n\pi x) $$
why are the following three cases the way they are? $$q_0 = 0$$$$ q_1 = \cosh(\pi y )$$$$q_n = 0, n\geq 2 $$
how do I determine those, do I plug $n=0,1,2$ in?
for instance for $n=0$
$$\cosh(\pi y )\cos(\pi x) = q_0(y) \ $$
but that doesn't give $q_0 = 0 $
The summation means $$ \sum_{n=0}^\infty q_n(y)\cos(n\pi x)=q_0(y)+q_1(y)\cos(\pi x)+q_2(y)\cos(2\pi x)+\ldots. $$ The only possibility to get a term identical to $\cosh(\pi y)\cos(\pi x)$ is setting all to 0 except for $n=1$. Only then you choose $q_1(y)=\cosh(\pi y)$ and you are done.