Error bound for the asymptotic expansion of the Hartman-Watson integral

This note gives a bound on the error of the leading term of the $t\to 0$ asymptotic expansion of the Hartman-Watson distribution $\theta (r,t)$ in the regime $rt=\rho$ constant. The leading order term has the form $\theta (\rho \mathrm{/}t,t)=\frac{1}{2\pi t}{e}^{-\frac{1}{t}(F(\rho )-{\pi }^2\mathrm{/}2)}G(\rho )(1+\vartheta (t,\rho ))$, where the error term is bounded uniformly over $\rho$ as $\mathrm{\mid }\vartheta (t,\rho )\mathrm{\mid }\le \frac{1}{70}t$.