# Bessel potential

In mathematics, the **Bessel potential** is a potential (named after Friedrich Wilhelm Bessel) similar to the **Riesz potential** but with better decay properties at infinity.

If *s* is a complex number with positive real part then the Bessel potential of order *s* is the operator

where Δ is the Laplace operator and the fractional power is defined using Fourier transforms.

Yukawa potentials are particular cases of Bessel potentials for in the 3-dimensional space.

## Representation in Fourier space

The Bessel potential acts by multiplication on the Fourier transforms: for each

## Integral representations

When , the Bessel potential on can be represented by

where the Bessel kernel is defined for by the integral formula ^{[1]}

Here denotes the Gamma function.
The Bessel kernel can also be represented for by^{[2]}

## Asymptotics

At the origin, one has as ,^{[3]}

In particular, when the Bessel potential behaves asymptotically as the Riesz potential.

At infinity, one has, as , ^{[4]}

## See also

## References

- ↑ Stein, Elias (1970).
*Singular integrals and differentiability properties of functions*. Princeton University Press. Chapter V eq. (26). ISBN 0-691-08079-8. - ↑ N. Aronszajn; K. T. Smith (1961). "Theory of Bessel potentials I".
*Ann. Inst. Fourier*.**11**. 385–475, (4,2). - ↑ N. Aronszajn; K. T. Smith (1961). "Theory of Bessel potentials I".
*Ann. Inst. Fourier*.**11**. 385–475, (4,3). - ↑ N. Aronszajn; K. T. Smith (1961). "Theory of Bessel potentials I".
*Ann. Inst. Fourier*.**11**: 385–475.

- Duduchava, R. (2001), "Bessel potential operator", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4 - Grafakos, Loukas (2009),
*Modern Fourier analysis*, Graduate Texts in Mathematics,**250**(2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-09434-2, ISBN 978-0-387-09433-5, MR 2463316 - Hedberg, L.I. (2001), "Bessel potential space", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4 - Solomentsev, E.D. (2001), "B/b015870", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4 - Stein, Elias (1970),
*Singular integrals and differentiability properties of functions*, Princeton, NJ: Princeton University Press, ISBN 0-691-08079-8