Kontorovich–Lebedev transform

Summary

The Kontorovich–Lebedev transform integrates a function $f$ on $(0, \infty)$ against the modified Bessel function $K_{iy}(x)$ of imaginary order $iy$. Unlike most transforms, the transform variable $y$ enters through the order of the Bessel function, not through its argument. It has an explicit inversion formula with the weight $y \sinh(\pi y)$. It is used to solve boundary value problems in wedge-shaped and conical regions, where the separation of variables leads to Bessel functions of imaginary order.