In this paper we provide a unified approach to a family of integrals of Mellin--Barnes type using distribution theory and Fourier transforms. Interesting features arise in many of the cases which call for the application of pull-backs of distributi
ons via smooth submersive maps defined by Hormander. We derive by this method the integrals of Hecke and Sonine relating to various types of Bessel functions which have found applications in analytic and algebraic number theory.
A simple proof of Ramanujans formula for the Fourier transform of the square of the modulus of the Gamma function restricted to a vertical line in the right half-plane is given. The result is extended to vertical lines in the left half-plane by sol
ving an inhomogeneous ODE. We then use it to calculate the jump across the imaginary axis.
