The Parabolic Mellin Transform: Gamma and Zeta Integral Representations

Peter Reinhard Hansen, Chen Tong

We introduce the Parabolic Mellin Transform (PMT), defined by $P_{\sigma }[f](z)={\int }_{-\infty }^{\infty }{w}^{2z}f(w^2)dt$, where $w=\sigma +it$ and $\sigma >0$. Under the substitution $u=w^2$, the ver…