The electrostatic potential of a uniformly charged disk as the source of novel mathematical identities

We use two different frameworks to calculate the electrostatic potential created by a uniformly charged disk in its plane. We first implement a standard direct integration approach and carry out the calculations to the extent possible while keeping them simple. We then apply a different method that relies on Bessel function transformations. This method allows us to derive compact analytic expressions for the electrostatic potential in terms of complete elliptic integrals of the first and second kind. By comparing the results obtained through the application of the two different approaches, we uncover interesting transformation formulas and novel mathematical identities that might be of interest to a broad range of researchers in mathematical and physical sciences.