View source for Lebedev transform

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The integral transform

$$F(\tau)=\int\limits_0^\infty[I_{i\tau}(x)+I_{-i\tau}(x)]K_{i\tau}(x)f(x)dx,\quad0\leq\tau<\infty,$$

where $I_\nu(x)$ and $K_\nu(x)$ are the modified [[Cylinder functions|cylinder functions]]. It was introduced by N.N. Lebedev [[#References|[1]]]. If

$$x^{-1/2}f(x)\in L(0,1),\quad x^{1/2}f(x)\in L(1,\infty),$$

then for almost-all $x$ one has the inversion formula

$$f(x)=-\frac{4}{\pi^2}\int\limits_0^\infty F(\tau)\tau\sinh\pi\tau K_{i\tau}^2(x)d\tau.$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Lebedev,   "On an integral representation of an arbitrary function in terms of squares of MacDonald functions with imaginary index"  ''Sibirsk. Mat. Zh.'' , '''3''' :  2  (1962)  pp. 213–222  (In Russian)</TD></TR></table>



====Comments====
The following transform pair is also called a Lebedev transform (or [[Kontorovich-Lebedev-transform(2)|Kontorovich–Lebedev transform]])

$$G(\tau)=\int\limits_0^\infty g(x)x^{-1/2}K_{i\tau}(x)dx,$$

$$g(x)=\frac{2}{\pi^2}\frac{1}{\sqrt x}\int\limits_0^\infty\tau\sinh\pi\tau K_{i\tau}(x)G(\tau)d\tau.$$

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.N. Lebedev,   "Special functions and their applications" , Prentice-Hall  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.N. Sneddon,   "The use of integral transforms" , McGraw-Hill  (1972)  pp. Chapt. 6</TD></TR></table>