An Integral representation of R(s) due to Gabcke

Gabcke proved a new integral expression for the auxiliary Riemann function \[{ R}(s)=2s/2πs/2eπ i(s-1)/4∫₋₁₂₁₂ {e-π i u^2/2+π i u}{2icosπ u}U(s-12,√2π i/4u)\,du,\] where U(ν,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 {equation}{ R}(s)=-2^s πs/2eπ i s/4∫-∞^∞ {e-π x^2H₋ₛ(x√π)}{1+e-2πω x}\,dx.{equation} where H_ν(z) is the generalized Hermite polynomial.