BesselJ

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[J, ν])(z) is an analytical function of ν and z which is defined over (ℂ^2).

Symmetries and periodicities

Parity

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

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

With respect to ν

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

Branch cuts

With respect to z

With respect to ν

For fixed z, the function (Subscript[J, 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 inside Stokes sectors

Expansions for any z  in exponential form

Expansions for any z  in trigonometric form

Residue representations

Integral representations

On the real axis

Of the direct function

Contour integral representations

Integral representations of negative integer order

Limit representations

Generating functions

P. Abbott

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

Relations of special kind

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

Definite integration

For the direct function itself

Involving the direct function

Integral transforms

Fourier exp transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Mellin transforms

Hankel transforms

Summation

Infinite summation

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric0F1Regularized

Involving Hypergeometric0F1

Involving Hypergeometric1F1

Through Meijer G

Classical cases for the direct function itself

Classical cases involving Cos

Classical cases involving Sin

Classical cases involving Cos, Sin

Classical cases for powers of Bessel BesselJ

Classical cases for products of Bessel BesselJ

Classical cases involving Bessel BesselI

Classical cases involving Bessel BesselK

Classical cases involving Bessel BesselY

Classical cases involving Cos, Sin, BesselY

Classical cases involving Hypergeometric0F1

Classical cases involving Hypergeometric0F1Regularized

Generalized cases for the direct function itself

Generalized cases involving Cos

Generalized cases involving Sin

Generalized cases involving Cos, Sin

Generalized cases for powers of Bessel BesselJ

Generalized cases for products of Bessel BesselJ

Generalized cases involving Bessel BesselI

Generalized cases involving Bessel BesselK

Generalized cases involving Bessel BesselY

Generalized cases involving Cos, Sin, BesselY

Generalized cases involving Hypergeometric0F1

Generalized cases involving Hypergeometric0F1Regularized

Through other functions

Representations through equivalent functions

With related functions

Inequalities

Zeros

When ν is real, the functions (Subscript[J, ν])(z)and (((∂(Subscript[J, ν])(z))/(∂z))) each have an infinite number of real zeros, all of which are simple with the possible exception of z= 0.

When ν≥-1 , the zeros of (Subscript[J, ν])(z) are all real. If ν< -1 and ν is not an integer, the number of complex zeros of (Subscript[J, ν])(z) is  2⌊-ν⌋; if ⌊-ν⌋ is odd, two of these zeros lie on the imaginary axis.

If ν>0, all zeros of (((∂(Subscript[J, ν])(z))/(∂z))) are real.

Theorems

Hankel transformation

(OverHat[(Subscript[f,  ν])])(y)==(∫_0^∞)f(x)(Sqrt[x y])(Subscript[J,  ν])(x y)dx⇔ f(x)==(∫_0^∞)(OverHat[(Subscript[f,  ν])])(y)(Sqrt[x y])(Subscript[J,  ν])(x y)dy/;Re(ν)≥-(1/2).

Green's function for the Klein-Gordon equation

Green's function for the Klein-Gordon equation (□+(m^2))G==δ(x-0)δ(t) is given by

G(x,t,0,0)==((θ(t))/(2π))δ((t^2)-|x|2)-(m/(2π))θ(t-|x|)(((Subscript[J, 1])(m(Sqrt[(t^2)-|x|2])))/(m(Sqrt[(t^2)-|x|2]))).

The initial value problem for a periodic chain of coupled harmonic oscillators

The initial value problem for a periodic chain of coupled harmonic oscillators with Hamiltonian H==(∑_(k=1)^n)(1/2)((Subscript[p, j]^2)+((((Subscript[q, j])-(Subscript[q, j+1])))^2)) with periodic boundary conditions can be expressed as (Subscript[ξ, n])(t)==(∑_(k=-∞)^∞)(Subscript[ξ, k])(0)(Subscript[J, k-n])(t) where (Subscript[ξ, 2n])(t)==(Subscript[p, n])(t) and (Subscript[ξ, 2n+1])(t)==(Subscript[q, n+1])(t)-(Subscript[q, n])(t).

The solution of the Kepler equation

The solution of the Kepler equation w(z)-ε sin(w(z))==z is w(z)==z+2(∑_(k=1)^∞)((sin(k z))/k)(Subscript[J, k])(k ε)/;0<ε<1.

The potential of a flat circular disk

The potential V(r,z) of a flat circular disk of radius 1 in cylindrical coordinates is given by V(r,z)∝(2/π)(∫_o^∞)((sin(ξ))/ξ)(Subscript[J, 0])(r ξ)(e^(-ξ|z|))dξ.

The pair correlation function for two electrons

The pair correlation function g(r) for two electrons in the plane wave approximation is given by g(r)==1-((9π)/4)((((Subscript[J, 3/2])((Subscript[k, F])r))^2)/((((Subscript[k, F])r))^3)) where (Subscript[k, F]) is the Fermi wave length.

A solution to the Toda equation

A solution to the Toda equation (∂^2)((Θ(x,n,t))/(∂(t^2)))+(∂^2)((Θ(x,n,t))/(∂(x^2)))==2((e^(2(Θ(x,n+1,t)-Θ(x,n,t))))-(e^(2(Θ(x,n,t)-Θ(x,n-1,t))))) is given by

Θ(x,n,t)==(1/2)log(1+(((a^2)(Subscript[J, n+1]^2)(2(Sqrt[(t^2)+(x^2)])))/(1-(a^2)(1+(∑_(k=n+1)^∞)(Subscript[J, k]^2)(2(Sqrt[(t^2)+(x^2)]))))))/;(a^2)<(1/4).

The components of the magnetic field in a cylindrical coordinate

Let (Subscript[B, z]) , (Subscript[B, ϕ]), (Subscript[B, r]) the components of the magnetic field in a cylindrical coordinate system {r,ϕ,z} such that B is a force-free magnetic field (that is, div B = 0 and curl B= a B hold simultaneously). One such solution is

(Subscript[B, z])=ξ (e^(-l z))  (Subscript[J, 0])(r ξ), (Subscript[B, ϕ])=a (e^(-l z))  (Subscript[J, 1])(r ξ),(Subscript[B, r])=l (e^(-l z))  (Subscript[J, 1])(r ξ)/;(ξ^2)==(a^2)+(l^2).

The propagator for the inverse square potential

The function G(x,t;y)==((Sqrt[x y])/t)exp(i (((x^2)+(y^2))/(2t)))(Subscript[J, λ+(1/2)])(((x y)/t)) is the propagator for the singular potential V(x)==((λ(λ+1))/2)(1/(x^2))and satisfies

i(∂/(∂t))G(x,t;y)==-(1/2)((∂^2)/(∂(x^2)))G(x,t;y)+V(x)G(x,t;y)∧G(x,0;y)==δ(x-y).

The solution of the 1+1 dimensional Dirac equation

The spinor ψ(x,t)=={i (((t+x))/(Sqrt[(t^2)-(x^2)])) (Subscript[J, 1])(((m t)/γ)),(Subscript[J, 0])(((m t)/γ))} , γ==1/(Sqrt[1-((x^2)/(t^2))])is a solution of the 1+1 dimensional Dirac equation

i ((∂ψ(x,t))/(∂t))==-i (({{1, 0}, {0, -1}}))·((∂ψ(x,t))/(∂x))-m (({{0, 1}, {1, 0}}))·ψ(x,t).

Vibrating circular membrane

Eigenfunctions

The displacement u("r",t) of a vibrating membrane, at each point "r"∈(ℝ^2) as a function of t, satisfies the wave equation

Δ u("r",t)-(1/(Subscript[c, "ph"]^2))(((∂^2)u("r",t))/(∂(t^2)))=0.

Here, Δ is the Laplace operator and (Subscript[c, "ph"]^2) (= tension per unit surface density) is the phase velocity. For a circular membrane of radius (Subscript[r, 0]) that is fixed at the boundary and assumimg harmonic time dependence u("r",t)=cos(ωt+α)u("r")yields in a polar coordinate system

(Subscript[u, m,k])("r")=cos(m φ)(Subscript[J, |m|])((Sqrt[(Subscript[λ, m,k])])r)
 (Subscript[u, m,k])("r")=sin(mφ)(Subscript[J, |m|])((Sqrt[(Subscript[λ, m,k])])r)

(Subscript[λ, m,k])=(Subscript[μ, "m,k"]^2)/(Subscript[r, 0]^2) where (Subscript[μ, "m,k"]^2) is the kth zero of the Bessel function (Subscript[J, m])(z).

History

– J. Bernoulli (1703); D. Bernoulli (1733); 
– F. W. Bessel (1816,1824)
- O. Schlömilch (1857) used the name "Bessel functions"
– E. Lommel (1868) used arbitrary real indices
– H. Hankel (1869) used arbitrary complex indices
– I. Todhunter (1875); N. J. Sonin (1880) derived the recurrence relations

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