# Kontorovich-Lebedev-transform(2)

The integral transform

$$F ( \tau ) = \ \int\limits _ { 0 } ^ \infty K _ {i \tau } ( x) f ( x) d x ,$$

where $K _ \nu$ is the Macdonald function.

If $f$ is of bounded variation in a neighbourhood of a point $x = x _ {0} > 0$ and if

$$f ( x) \mathop{\rm ln} x \in L \left ( 0 , \frac{1}{2} \right ) ,\ \ f ( x) \sqrt x \in L \left ( \frac{1}{2} , \infty \right ) ,$$

then the following inversion formula holds:

$$\frac{f ( x _ {0} + ) + f ( x _ {0} - ) }{2 } =$$

$$= \ \frac{2}{\pi ^ {2} x _ {0} } \int\limits _ { 0 } ^ \infty K _ {i \tau } ( x _ {0} ) \tau \sinh \pi \tau F ( \tau ) d \tau .$$

Let $f _ {i}$, $i = 1 , 2$, be real-valued functions with

$$f _ {i} ( x) x ^ {-3/4} \in L ( 0 , \infty ) ,\ \ f _ {i} ( x) \in L _ {2} ( 0 , \infty ) ;$$

and let

$$F _ {i} ( \tau ) = \ \int\limits _ { 0 } ^ \infty \frac{\sqrt {2 \tau \sinh \pi \tau } } \pi \frac{K _ {i \tau } }{\sqrt x } f _ {i} ( x) d x .$$

Then

$$\int\limits _ { 0 } ^ \infty F _ {1} ( \tau ) F _ {2} ( \tau ) d \tau = \ \int\limits _ { 0 } ^ \infty f _ {1} ( x ) f _ {2} ( x) d x$$

(Parseval's identity).

The finite Kontorovich–Lebedev transform has the form

$$F ( \tau ) = \ \frac{2 \pi \sinh \pi \tau }{\pi ^ {2} | I _ {i \alpha } ( \alpha ) | ^ {2} } \times$$

$$\times \int\limits _ { 0 } ^ \alpha [ K _ {i \tau } ( \alpha ) I _ {i \tau } ( x) - I _ {i \tau } ( \alpha ) K _ {i \tau } ( x) ] f ( x) \frac{dx}{x} ,$$

$\tau > 0$, where $I _ \nu$ is the modified Bessel function (see ).

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

How to Cite This Entry:
Kontorovich-Lebedev-transform(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kontorovich-Lebedev-transform(2)&oldid=52090
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article