= Fourier series of a periodically extended exponential
{c}
{title2=$f(x)=e^{\kappa x},\quad -L<x<L$}
Let $L>0$ and real $\kappa\ne0$. Extend $e^{\kappa x}$ with period $2L$. In the <Fourier series> convention $a_0/2+\sum_{n\geq1}[a_n\cos(n\pi x/L)+b_n\sin(n\pi x/L)]$, put $q_n=n\pi/L$. Direct <integration> gives
$$
a_0=\frac{2\sinh(\kappa L)}{\kappa L},\qquad
a_n=\frac{2\kappa(-1)^n\sinh(\kappa L)}{L(\kappa^2+q_n^2)},\qquad
b_n=-\frac{2q_n(-1)^n\sinh(\kappa L)}{L(\kappa^2+q_n^2)}.
$$
For example, the required <antiderivatives> are $e^{\kappa x}(\kappa\cos q_nx+q_n\sin q_nx)/(\kappa^2+q_n^2)$ and $e^{\kappa x}(\kappa\sin q_nx-q_n\cos q_nx)/(\kappa^2+q_n^2)$. The <Dirichlet-Jordan convergence theorem> gives the exponential value away from the jumps and their midpoint $\cosh(\kappa L)$ at odd multiples of $L$. For $\kappa=0$, the <function> is identically one, with $a_0=2$ and all other <Fourier coefficients> zero.
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