Inmathematics, theKontorovich–Lebedev transform is anintegral transform which uses a Macdonald function (modifiedBessel function of the second kind) withimaginary index as itskernel. Unlike other Bessel function transforms, such as theHankel transform, this transform involves integrating over theindex of the function rather than its argument.

The transform of a function${\displaystyle f(x)}$ and its inverse (provided they exist) are given below:

${\displaystyle g(y)={\int }_0^{\mathrm{\infty }}f(x)K_{iy}(x)dx}$

${\displaystyle f(x)=\frac{2}{{\pi }^{2}x}{\int }_0^{\mathrm{\infty }}g(y)K_{iy}(x)\sinh (\pi y)ydy.}$

Laguerre previously studied a similar transform regardingLaguerre function as:

${\displaystyle g(y)={\int }_0^{\mathrm{\infty }}f(x)e^{-x}{L}_y(x)dx}$

${\displaystyle f(x)={\int }_0^{\mathrm{\infty }}\frac{g(y)}{\mathrm{\Gamma }(y)}{L}_y(x)dy.}$

Erdélyiet al., for instance, contains a short list of Kontorovich–Lebedev transforms as well references to the original work of Kontorovich and Lebedev in the late 1930s. This transform is mostly used in solving theLaplace equation incylindrical coordinates for wedge shaped domains by the method ofseparation of variables.

• A. Erdélyiet al.Table of Integral Transforms Vol. 2 (McGraw Hill 1954)
• I.N. Sneddon,The use of integral Transforms, (McGraw Hill, New York 1972)
• "Kontorovich–Lebedev transform",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

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