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 spaceEdit

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

 

Integral representationsEdit

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]

 

AsymptoticsEdit

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 alsoEdit

ReferencesEdit

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