Bessel potential

In mathematics, the Bessel potential is a potential 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
Here denotes the Gamma function.
The Bessel kernel can also be represented for by
This last expression can be more succinctly written in terms of a modified Bessel function, for which the potential gets its name:

Asymptotics

At the origin, one has as,
In particular, when the Bessel potential behaves asymptotically as the Riesz potential.
At infinity, one has, as,