Bessel potential

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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[edit]

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

Integral representations[edit]

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[edit]

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[edit]

References[edit]

  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.