= Fourier series of periodic hyperbolic sine and cosine
{c}
For $\beta>0$, periodically extend $\cosh(\beta x)$ and $\sinh(\beta x)$ from $-1<x\leq1$ with period $2$. Their <Fourier series> are
$$
\cosh(\beta x)=\frac{\sinh\beta}{\beta}+2\beta\sinh\beta\sum_{n=1}^{\infty}\frac{(-1)^n\cos(n\pi x)}{\beta^2+n^2\pi^2},
$$
and, away from odd integers,
$$
\sinh(\beta x)=2\pi\sinh\beta\sum_{n=1}^{\infty}\frac{(-1)^{n+1}n\sin(n\pi x)}{\beta^2+n^2\pi^2}.
$$
The first identity follows by <integration> of $\cosh(\beta x)\cos(n\pi x)$ on $[-1,1]$, using the <antiderivative>
$$
\frac{\beta\sinh(\beta x)\cos(n\pi x)+n\pi\cosh(\beta x)\sin(n\pi x)}{\beta^2+n^2\pi^2}.
$$
For the second identity, <integration by parts> and $d\cosh(\beta x)/dx=\beta\sinh(\beta x)$ show that its sine coefficient is $-n\pi/\beta$ times the first series' cosine coefficient. This avoids assuming that a <Fourier series> can always be differentiated term by term. The periodic <hyperbolic cosine> is continuous, so its series converges to it everywhere. The periodic <hyperbolic sine> has a <jump discontinuity> at each odd integer: the <Dirichlet-Jordan convergence theorem> gives the average of its two limiting values, namely zero, there. Both identities hold in the sense of <mean-square convergence>, which suffices for <Parseval's identity>.
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