I'm working on some problems with relation to elasticity (plate mechanics in specific) and while I've made some progress the following sum is giving me a hard time
$\sum_{m=0}^{\infty}\frac{(-1)^m}{(2m+1)}\exp\left(-(2m+1)^2t\right)$
where $t\geq 0$. There are a number of special functions that comes close, e.g. the (second) Jacobi Theta functions, and there could be some relation to arctanh, but I cannot really find a closed form solution. Any help or hints would be greatly appreciated.
Here is a representation in terms of Jacobi theta functions. We start from the definition of $\displaystyle \vartheta_3 (z,q)=1+2\sum_{n=1}^\infty q^{n^2}\cos(2 nz),$ the same as that of Mathematica. If we integrate from $z=0$ to $\pi/4$, we obtain $\displaystyle \int_0^{\pi/4} \vartheta_3 (z,q)\,dz=\frac{\pi}{4}+\sum_{n=1}^\infty \frac{1}{n} q^{n^2}\sin(\frac{\pi n}{2})$. But the sinuosoid vanishes if $n$ is even, and alternates sign for odd $n$. Consequently, by taking $q=e^{-t}$ and rearranging slightly we have the Jacobi-theta representation
$$\boxed{\displaystyle \int_0^{\pi/4} \left(\vartheta_3 (z,e^{-t})-1\right)\,dz=\sum_{m=0}^\infty \frac{(-1)^m}{2m+1} e^{-(2m+1)^2}}$$