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	~~The writer is known by~~ the ~~title of Numbers Wunder. She~~ is a ~~librarian but she~~'~~s usually needed her personal company~~. ~~For a while she~~'~~s been in South Dakota~~. ~~To do aerobics is~~ a ~~thing that I~~'~~m totally addicted to.~~<br><br>~~Also visit my homepage~~ :: ~~[http~~://~~youulike~~.~~com/blogs/114396/390615/items-~~to-~~know~~-~~when~~-~~confronted~~-~~wi over~~ the ~~counter std test~~]		{{Orphan\|date=February 2013}}

			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]]


			{{Mathanalysis-stub}}

en>SmackBot: remove Erik9bot category,outdated, tag and general fixes

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remove Erik9bot category,outdated, tag and general fixes

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The writer is known by the title of Numbers Wunder. She is a librarian but she's usually needed her personal company. For a while she's been in South Dakota. To do aerobics is a thing that I'm totally addicted to.<br><br>Also visit my homepage :: [http://youulike.com/blogs/114396/390615/items-to-know-when-confronted-wi over the counter std test]

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