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. 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 {displaystyle s=2} in the 3-dimensional space. The Bessel potential acts by multiplication on the Fourier transforms: for each ξ ∈ R d {displaystyle xi in mathbb {R} ^{d}} When s > 0 {displaystyle s>0} , the Bessel potential on R d {displaystyle mathbb {R} ^{d}} can be represented by where the Bessel kernel G s {displaystyle G_{s}} is defined for x ∈ R d ∖ { 0 } {displaystyle xin mathbb {R} ^{d}setminus {0}} by the integral formula Here Γ {displaystyle Gamma } denotes the Gamma function.The Bessel kernel can also be represented for x ∈ R d ∖ { 0 } {displaystyle xin mathbb {R} ^{d}setminus {0}} by At the origin, one has as | x | → 0 {displaystyle vert xvert o 0} ,