SphericalBesselJ

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  BAD

Limit representations

Generating functions

P. Abbott

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

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 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 SphericalBesselJ

Classical cases for products of SphericalBesselJ

Classical cases involving Bessel BesselJ

Classical cases involving Bessel BesselY

Classical cases involving BesselJ, BesselY, SphericalBesselY

Classical cases involving Bessel BesselI

Classical cases involving Bessel BesselK

Classical cases involving SphericalBesselY

Classical cases involving Cos, Sin, SphericalBesselY

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 SphericalesselJ

Generalized cases for products of SphericalBesselJ

Generalized cases involving Bessel BesselJ

Generalized cases involving Bessel BesselY

Generalized cases involving BesselJ, BesselY, SphericalBesselY

Generalized cases involving Bessel BesselI

Generalized cases involving Bessel BesselK

Generalized cases involving SphericalBesselY

Generalized cases involving Cos, Sin, SphericalBesselY

Generalized cases involving Hypergeometric0F1

Generalized cases involving Hypergeometric0F1Regularized

Through other functions

Representations through equivalent functions

With related functions

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