BesselK

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed ν

For fixed z

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

(Subscript[K, ν])(z) is an analytical function of ν and z which is defined in (ℂ^2).

Symmetries and periodicities

Parity

(Subscript[K, ν])(z) is an even function with respect to its parameter.

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

For fixed ν, the function (Subscript[K, ν])(z) has an essential singularity at z==(OverTilde[∞]). At the same time, the point z==(OverTilde[∞]) is a branch point.

With respect to ν

For fixed z, the function (Subscript[K, ν])(z) has only one singular point at ν =(OverTilde[∞]).  It is an essential singular point.

Branch points

With respect to z

With respect to ν

For fixed z, the function (Subscript[K, ν])(z) does not have branch points.

Branch cuts

With respect to z

With respect to ν

For fixed z, the function (Subscript[K, ν])(z) is an entire function of ν and does not have branch cuts.

Series representations

Generalized power series

Expansions at ν==±n

Expansions at generic point z==(Subscript[z, 0])

Expansions on branch cuts

Expansions at z==0

Asymptotic series expansions

Expansions for any z  in exponential form

Expansions at ν==i ∞

Residue representations

Integral representations

On the real axis

Of the direct function

Contour integral representations

Limit representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Involving related functions

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Addition formulas

Multiple arguments

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations between contiguous functions

Differentiation

Low-order differentiation

With respect to ν

Brychkov Yu.A. (2005)

Brychkov Yu.A. (2005)

With respect to z

Symbolic differentiation

With respect to ν

With respect to z

Fractional integro-differentiation

With respect to z

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Involving functions of the direct function

Involving functions of the direct function and elementary functions

Involving direct function and Bessel-type functions

Definite integration

For the direct function itself

Involving the direct function

Integral transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Mellin transforms

Hankel transforms

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric0F1Regularized

Involving Hypergeometric0F1

Involving hypergeometric HypergeometricU

Involving Hypergeometric1F1

Through Meijer G

Classical cases for the direct function itself

Classical cases involving Exp

Classical cases involving Cosh

Classical cases involving Sinh

Classical cases for powers of Bessel BesselK

Classical cases for products of Bessel BesselK

Classical cases involving Bessel BesselJ

Classical cases involving Bessel BesselI

Classical cases involving Bessel BesselY

Classical cases involving Hypergeometric0F1

Classical cases involving Hypergeometric0F1Regularized

Generalized cases for the direct function itself

Generalized cases involving Cosh

Generalized cases involving Sinh

Generalized cases involving AiryAi

Generalized cases involving AiryAiPrime

Generalized cases involving AiryBi

Generalized cases involving AiryBiPrime

Generalized cases for powers of Bessel BesselK

Generalized cases for products of Bessel BesselK

Generalized cases involving Bessel BesselJ

Generalized cases involving Bessel BesselI

Generalized cases involving Bessel BesselY

Generalized cases involving Hypergeometric0F1

Generalized cases involving Hypergeometric0F1Regularized

Representations through equivalent functions

With related functions

Zeros

The function (Subscript[K, ν])(z) has no zeros in the region |Arg(z)|≤(π/2) for any real ν.

Theorems

Kontorovich-Lebedev transformation

(Overscript[f, ⋀])(y)==(∫_0^∞)f(x)(Subscript[K,  i y])(x)dx⇔f(x)==(2/((π^2)x ))(∫_0^∞)(Overscript[f, ⋀])(y)y sinh(π y)(Subscript[K,  i y])(x)dy.

Meijer transformation

For Re(ν)≥-(1/(2 )) the following identity holds:

(Overscript[(Subscript[f,  ν]), ⋀])(y)==(∫_0^∞)f(x)(Sqrt[x y])(Subscript[K,  ν])(x y)dx ⇔ f(x)==(1/(π i ))(∫_(γ-i ∞)^(γ+i ∞))(Overscript[(Subscript[f,  ν]), ⋀])(y)(Sqrt[x y])(Subscript[I,  ν])(x y)dy

The Green's function of the time independent Schrödinger equation

The Green's function G(x',x;ε) of the time–independent Schrödinger equation for a free particle in d dimensions -Δ G(x',x;ε)-ε G(x',x;ε)==-i δ(x'-x) is given by

G(x',x;ε)==-((i ((((Sqrt[-ε])))^(d-2)))/(((2π))^(d/2))) (((Subscript[K, d/2-1])((Sqrt[-ε])|x'-x|))/((((Sqrt[-ε])|x'-x|))^(d/2-1))).

The Newton-Wigner wave function

The Newton-Wigner wave function (Subscript[ψ, NW])(t,x) is related to the covariant wave function ψ(t,x) by (Subscript[ψ, NW])(t,x)==∫(Sqrt[(π/2)])(((((2(Subscript[λ, C]))/(|x-y|))))^(5/4))(Subscript[K, (5/4)])(((|x-y|)/(Subscript[λ, C]))) ψ(t,y)dy,  where (Subscript[λ, C]) is the Compton wavelength.

Green's function of the Helmholtz operator

Green's function of the Helmholtz operator in the xy–plane ((∂_(x x))+(∂_(y y))+(k^2))G(x,y)==δ(x)δ(y) : G(x,y)==-(1/(2π))(Subscript[K, 1])(-i k (Sqrt[(x^2)+(y^2)])).

History

– H. M. MacDonald (1899)

Copyright