Mellin transform method

Summary

The Mellin transform method solves integral equations on $(0, \infty)$ whose kernel depends only on the ratio $x / t$ of the variables. Such an integral is a Mellin convolution, and the Mellin transform turns it into a product. The integral equation becomes an algebraic equation in a vertical strip of the complex plane, and the Mellin inversion theorem brings the solution back. The method covers all kernels that are homogeneous of degree $-1$, such as $\frac{1}{x + t}$. Such kernels arise in problems with scaling symmetry, such as wedges, cones and cracks.