On the Kontorovich-Lebedev transform (Q2910113)

The authors consider the Kontorovich-Lebedev transform in the form NEWLINE\[NEWLINEg(t)= \int^\infty_0 f(x)\,K_{it}(x){dx\over x},NEWLINE\]NEWLINE where \(K_{it}\) is the Macdonald-Bessel function, and give a proof of the inversion formula. The proof is based on the evaluation of the integral NEWLINE\[NEWLINE\int^\infty_{-\infty} K_{it}(x)\,K_{it}(y)\, K_{it}(z){dt\over \Gamma(it)\Gamma(-it)},NEWLINE\]NEWLINE which is interesting by itself.