 Fourier series of a periodically extended exponential

Let $L>0$ and real $\kappa \ne 0$. Extend $e^{\kappa x}$ with period $2L$. In the Fourier series convention $a_0/2+{\sum }_{n\ge 1}[a_n\cos (n\pi x/L)+b_n\sin (n\pi x/L)]$, put $q_n=n\pi /L$. Direct integration givesFor example, the required antiderivatives are $e^{\kappa x}(\kappa \cos q_{n}x+q_n\sin q_{n}x)/({\kappa }^2+q_n^2)$ and $e^{\kappa x}(\kappa \sin q_{n}x-q_n\cos q_{n}x)/({\kappa }^2+q_n^2)$. 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.