Fourier series of the form $\sum_{k=1}^{\infty}\frac{\sin(kx)}{k^{2}+a^{2}}$

Question

I have encountered some Fourier series of the form : $$\sum_{k=1}^{\infty}\frac{\sin(kx)}{k^{2}+a^{2}}$$ and $$\sum_{k=1}^{\infty}\frac{\cos(kx)}{k^{2}+a^{2}}$$ What is the domain of convergence of such series, and what functions -if any- do they represent ?

Answer 1

Since $\sum_{k=1}^\infty 1/(k^2+a^2)$ converges, these series converge for all real $x$. According to Maple,

$$ \sum_{k=1}^\infty\frac{\cos(kx)}{k^2+a^2} = {\frac {-i \left( {\Phi} \left( {{\rm e}^{ix}},1,-ia \right) a-{\Phi} \left( {{\rm e}^{ix}},1,ia \right) a+{\Phi} \left( {{\rm e}^{-ix}},1,-ia \right) a-{\Phi} \left( {{\rm e}^{-ix}},1,ia \right) a-4\,i \right) }{4{a}^{2}}}$$ $$\sum_{k=1}^\infty\frac{\sin(kx)}{k^2+a^2} = {\frac {{\Phi} \left( {{\rm e}^{-ix}},1,-ia \right) -{ \Phi} \left( {{\rm e}^{-ix}},1,ia \right) -{\Phi} \left( {{\rm e}^{ix}},1,-ia \right) +{\Phi} \left( {{\rm e}^{ ix}},1,ia \right) }{4a}} $$ where $\Phi$ is the Lerch Phi function.