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 s=2 in the 3-dimensional space.

Representation in Fourier space

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

Integral representations

When s > 0, the Bessel potential on can be represented by where the Bessel kernel G_s is defined for by the integral formula Here \Gamma 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 0 < s < d the Bessel potential behaves asymptotically as the Riesz potential. At infinity, one has, as ,