Asymptotic expansion of the Mordell integral

Question

my question concerns the Mordell integral $$h(z;\tau):=\int_{-\infty}^\infty \frac{e^{\pi i\tau w^2-2\pi zw}}{\cosh(\pi w)}dw,\qquad \Im(\tau)>0,\quad z\in\mathbb{C},$$ which frequently occurs in the theory of mock modular forms.

I wonder whether one knows anything about the behaviour of this function for $\tau\rightarrow 0$. To be more precise, I am interested in the difference $$ h(3z-\tau;3\tau)-h(3z+\tau;3\tau)$$ where $z$ is real. One stumbles over this differnce when analyzing the modular transformation behaviour of the generating function of partition ranks.

Thanks a lot in advance!

Answer 1

The Maple code

h := unapply(int(exp(I*Pi*tau*w^2-2*Pi*z*w)/cosh(Pi*w), w = -infinity .. infinity), z, tau):
series(h(3*z-tau, 3*tau)-h(3*z+tau, 3*tau), tau, 2)

produces $$\left(\int_{-\infty}^\infty \frac {e^{-6\pi z w}(3i\pi w^2+2\pi w)} {\cosh (\pi w)}\,dw- \int_{-\infty}^\infty \frac {e^{-6\pi z w}(3i\pi w^2-2\pi w)} {\cosh (\pi w)}\,dw\right)\tau +O(\tau^2)=$$ $$\tau \int_{-\infty}^\infty \frac {8\pi\, w\, e^{-6\pi w z+\pi w}}{e^{2\pi w}+1}\,dw+O(\tau^2),\, \tau \to 0.$$