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In [[mathematics]], the '''Kontorovich–Lebedev transform''' is an [[integral transform]] which uses a Macdonald function (modified [[Bessel function]] of the second kind) with [[Imaginary number|imaginary]] index as its '''kernel'''. Unlike other Bessel function transforms, such as the [[Hankel transform]], this transform involves integrating over the '''index''' of the function rather than its argument.

The transform of a function ''&fnof;''(''x'') and its inverse (provided they exist) are given below:

:<math>g(y) = \int_0^\infty f(x) K_{iy}(x) \, dx  </math>

:<math>f(x) = \frac{2}{\pi^2 x} \int_0^\infty g(y) K_{iy}(x) \sinh (\pi y) y \, dy . </math>

Laguerre previously studied a similar transform regarding [[Laguerre function]] as:

:<math>g(y) = \int_0^\infty f(x)e^{-x} L_{y}(x) \, dx  </math>

:<math>f(x) = \int_0^\infty \frac{g(y)}{\Gamma (y)} L_y(x) \, dy.  </math>

Erdélyi ''et al.'', for instance, contains a short list of Kontorovich–Lebedev transforms as well references to the original work of Kontorovich and Lebedev<!-- I don't see any writings of Kontorovish and Lebedev cited here. --> in the late 1930s.  This transform is mostly used in solving the Laplace equation in cylindrical coordinates for wedge shaped domains by the method of separation of variables.

== References ==

* Erdélyi ''et al.'' ''Table of Integral Transforms Vol. 2'' (McGraw Hill 1954)
* I.N. Sneddon, ''The use of integral Transforms'', (McGraw Hill, New York 1972)
*{{Springer|id=k/k120090|title=Kontorovich–Lebedev transform}}

{{DEFAULTSORT:Kontorovich-Lebedev transform}}
[[Category:Integral transforms]]
[[Category:Special functions]]


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