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Mathematical potential From Wikipedia, the free encyclopedia
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.
The Bessel potential acts by multiplication on the Fourier transforms: for each
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]
This last expression can be more succinctly written in terms of a modified Bessel function,[3] for which the potential gets its name:
At the origin, one has as ,[4]
In particular, when the Bessel potential behaves asymptotically as the Riesz potential.
At infinity, one has, as , [5]
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