Evaluating a series of Gaussians and Sines

Question

I have derived an equation that includes the following sum: $$ \sum_{n=1}^\infty n \exp\left(-an^2\right) \sin\left(\frac{n \pi x }{L}\right). $$ Is there a way to figure out what function $f(x)$ this series is equal to? I also have a complementary equation with a similar sum given by: $$ \sum_{n=1}^\infty n (-1)^{n+1} \exp\left(-an^2\right) \sin\left(\frac{n \pi x }{L}\right), $$ for which I have the same question. My only thoughts are that I could go through Abramowitz and Stegun and look for some function that has this series (which I did spend some time doing), or that the above are Fourier Sine series. The latter fact immediately tells me that $$ \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n \pi x }{L}\right) dx = n \exp\left(-an^2\right), $$ but I don't see this helping. In general I am wondering the best way to look for the function that has a given series representation.

Answer 1

This is equal to

$$ \sum_{n=1}^{\infty} n e^{-n^2a}\sin\left(\frac{n\pi x}{L}\right)= -\frac{L}{\pi}\frac{{\rm d}}{{\rm d}x} \frac{\displaystyle\vartheta\left(\frac{x}{2L},\frac{ia}{\pi}\right)-1}{2} $$

where $\vartheta(z;\tau)$ is the Jacobi theta function

$$ \vartheta(z;\tau)=\sum_{-\infty}^{\infty} \exp(\pi in^2\tau+2\pi inz)=1+2\sum_{n=1}^{\infty} \exp(\pi in^2\tau)\cos(2\pi nz). $$

This isn't much different, but the special function has a name you can use to research.