Improper integral about exp appeared in Titchmarsh's book on the zeta function May I ask how to do the following integration?
$$\int_0^\infty \frac{e^{-(\pi n^{2}/x) -(\pi t^2 x)}}{\sqrt{x}} dx $$
where $t>0$, $n$ a positive integer.
This came up on page 32 (image) of Titchmarsh's book, The Theory of the Riemann Zeta-Function. Specifically for the sum involving $b_n$, I am wondering how to

multiply by $e^{-\pi t^{2} x}$ and integrate over $(0,\infty)$ 

 A: $$\begin{eqnarray*}
\int_0^\infty \frac{dx}{\sqrt{x}}\, \exp\left(-\frac{\pi n^2}{x} - \pi t^2 x\right)
&=& 2\sqrt{\frac{n}{t}} \int_0^\infty dz\, \exp\left(-\pi n t(z^2 + z^{-2})\right)
    \hspace{5ex} (\textrm{let } x=z^2 n/t) \\
&=& \sqrt{\frac{n}{t}} \int_{-\infty}^\infty ds\, e^{s/2}
    \exp\left(-2 \pi n t \cosh s\right)
    \hspace{5ex} (\textrm{let } z=e^{s/2}) \\
&=& 2\sqrt{\frac{n}{t}} \int_0^\infty ds\, \cosh\left(\frac{s}{2}\right) 
    \exp\left(-2 \pi n t \cosh s\right) \\
&=& 2\sqrt{\frac{n}{t}} K_{\frac{1}{2}} (2\pi n t)
    \hspace{5ex} (\textrm{modified Bessel function, 2nd kind}) \\
&=& \frac{e^{-2\pi n t}}{t}
\end{eqnarray*}$$
Addendum: An approach not involving special functions.
$$\begin{eqnarray*}
\int_0^\infty \frac{dx}{\sqrt{x}}\, 
    \exp\left(-\frac{\pi n^2}{x} - \pi t^2 x\right)
&=& 2\sqrt{\frac{n}{t}} \int_0^\infty dz\, 
    \exp\left(-\pi n t(z^2 + z^{-2})\right) 
    \hspace{5ex} (\textrm{as before}) \\
&=& 2\sqrt{\frac{n}{t}} \int_0^\infty dz\, 
    \exp\left(-\pi n t(z-z^{-1})^2 - 2\pi n t\right) \\
&=& \sqrt{\frac{n}{t}}  e^{-2\pi n t} \int_{-\infty}^\infty du\,
    \left(1+\frac{u}{\sqrt{u^2+4}}\right) e^{-\pi n t u^2} 
    \hspace{4ex} (z-z^{-1}=u) \\
&=& \frac{e^{-2\pi n t}}{t} 
    \hspace{5ex} (\textrm{odd integral vanishes; Gaussian left over}) 
\end{eqnarray*}$$
