An Integral representation of R (s) due to Gabcke

Gabcke proved a new integral expression for the auxiliary Riemann function \ R (s) =2^s/2^s/2e^ i (s-1) /4-₁₂₁₂ e^- i u²/2+ i u2i uU (s-12, 2^ i/4u) \, du, \ where U (, 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) =-2ˢ ^s/2e^ i s/4-^ e^- x²H-ₒ (x) 1+e^{-2 x}\, dx. equation where H_ (z) is the generalized Hermite polynomial.