Lebedev transform

The integral transform

where and are the modified cylinder functions. It was introduced by N.N. Lebedev [1]. If

then for almost-all one has the inversion formula

References

[1]	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)

Comments

The following transform pair is also called a Lebedev transform (or Kontorovich–Lebedev transform)

References

[a1]	N.N. Lebedev, "Special functions and their applications" , Prentice-Hall (1965) (Translated from Russian)
[a2]	I.N. Sneddon, "The use of integral transforms" , McGraw-Hill (1972) pp. Chapt. 6