Neumann-Weber integral transform for complex indices

Diffusion and heat equations are commonly investigated in mathematical physics and are solvable for potentials in polar coordinates with a separation into a radial and an angular equation. While the angular equation can be solved easily, a common method for solving the radial part consists in the application of the Neumann-Weber integral transform. The Neumann-Weber integral transform, however, has only been shown to be valid for real indices of Bessel functions. In this work, we generalize the Neumann-Weber transform to complex Bessel indices. The back transform then becomes dependent on zeros of Hankel functions, and we provide useful information for its numerical implementation. The results are relevant for solving diffusion equations and heat equations around cylindrical objects.