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The integral transform

where is the Macdonald function.

If is of bounded variation in a neighbourhood of a point and if

then the following inversion formula holds:

Let , , be real-valued functions with

and let


(Parseval's identity).

The finite Kontorovich–Lebedev transform has the form

, where is the modified Bessel function (see [3]).

The study of such transforms was initiated by M.I. Kontorovich and N.N. Lebedev (see [1], [2]).


[1] M.I. Kontorovich, N.N. Lebedev, "A method for the solution of problems in diffraction theory and related topics" Zh. Eksper. i. Toer. Fiz. , 8 : 10–11 (1938) pp. 1192–1206 (In Russian)
[2] N.N. Lebedev, Dokl. Akad. Nauk SSSR , 52 : 5 (1945) pp. 395–398
[3] Ya.S. Uflyand, E. Yushkova, Dokl. Akad. Nauk SSSR , 164 : 1 (1965) pp. 70–72
[4] V.A. Ditkin, A.P. Prudnikov, "Integral transforms and operational calculus" , Pergamon (1965) (Translated from Russian)


A transform table for the Kontorovich–Lebedev transform can be found in [a1]. A treatment in some detail of the transform is in [a2].


[a1] A. Erdelyi, W. Magnus, F. Oberhettinger, "Tables of integral transforms" , 1–2 , McGraw-Hill (1954) pp. Chapt. XII
[a2] I.N. Sneddon, "The use of integral transforms" , McGraw-Hill (1972) pp. Chapt. 6
How to Cite This Entry:
Kontorovich-Lebedev-transform(2). Yu.A. BrychkovA.P. Prudnikov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098