Inversion of the Kontorovich-Lebedev integral transform (Q1316842)

The Kontorovich-Lebedev integral transform is defined by \(K_ \lambda g=\int^ \infty_ 0K_{i\tau}(\lambda\rho)g(\rho)d\rho\), \(\tau\geq 0\), where \(K_ \nu(z)\) is the McDonald cylinder function and \(\lambda\) is a fixed complex number with \(\text{Re} \lambda>0\). Its inverse is \(K^{- 1}_ \lambda f={2\over\pi^ 2\rho}\int^ \infty_ 0K_{i\tau}(\lambda\rho)\tau sh \pi\tau f(\tau)d\tau\), \(\rho>0\). Results concerning a ``direct'' integral representation \((*)\) \(g(\rho)=(K_ \lambda^{-1}K_ \lambda g)(\rho)\), \(\rho>0\); giving the inverse of the Kantorovich-Lebedev transform, were obtained only in the case of real \(\lambda>0\). In the present paper we consider the problem concerning the conditions on function \(g(\rho)\), \(\rho>0\), under which the representation \((*)\) is valid when \(\text{Re} \lambda>0\).