I wonder if any of you knows how to find the value of the series $$\sum_{n=1}^{\infty}\frac{e^{-bn}}{n^2+z^{2}}.$$
This function shows up while solving a magnetostatic problem with complex-valued scalar potential.
Greetings
Oscar
I wonder if any of you knows how to find the value of the series $$\sum_{n=1}^{\infty}\frac{e^{-bn}}{n^2+z^{2}}.$$
This function shows up while solving a magnetostatic problem with complex-valued scalar potential.
Greetings
Oscar
The series has no expression in terms of elementary functions, but it does represent a special function (either the incomplete beta function $B$ or the Lerch transcendent $\Phi$): $$F(b,z)=\sum_{n=1}^\infty \frac{e^{-bn}}{n^2+z^2}=-\frac{1}{z}\,{\rm Im}\,\sum_{n=1}^\infty \frac{e^{-bn}}{n+iz}$$ $$\qquad=-\frac{1}{z}{\rm Im}\,e^{-b} \Phi (e^{-b},1,i z+1)=-\frac{1}{z}{\rm Im}\,e^{ibz}B_{e^{-b}}(i z+1,0).$$ (I'm assuming real $z$ and $b\geq 0$.) Two limits are: $$F(b,0)=\text{Li}_2\left(e^{-b}\right),$$ a polylog, while $$F(0,z)=\frac{\pi z \coth (\pi z)-1}{2 z^2}.$$