# 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 [3]).

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

#### References

 [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)