An Integral representation of $<mi\; mathvariant="script">R</mi>(s)$ due to Gabcke

Gabcke proved a new integral expression for the auxiliary Riemann function $$(s)=2^{s\mathrm{/}2}{\pi }^{s\mathrm{/}2}{e}^{\pi i(s-1)\mathrm{/}4}{\int }_{-\frac{1}{2}\searrow \frac{1}{2}}\frac{e^{-\pi i{u}^2\mathrm{/}2+\pi iu}}{2i\cos \pi u}U(s-\frac{1}{2},\sqrt{2\pi }{e}^{\pi i\mathrm{/}4}u)du,$$ where $U(\nu ,z)$ is the usual parabolic cylinder function. We give a new, shorter proof, which avoids the use of the Mordell integral. And we write it in the form \begin{equation}\mathop{\mathcal R}(s)=-2^s π^{s/2}e^{πi s/4}\int_{-\infty}^\infty \frac{e^{-πx^2}H_{-s}(x\sqrtπ)}{1+e^{-2πωx}},dx.\end{equation} where $H_{\nu }(z)$ is the generalized Hermite polynomial.