KelvinBer2

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[ber, ν])(z) is an analytical function of ν and z, which is defined in (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[ber, 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[ber, 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[ber, ν])(z) does not have branch points.

Branch cuts

With respect to z

With respect to ν

For fixed z, the function (Subscript[ber, 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

Residue representations

Integral representations

On the real axis

Of the direct function

Contour integral representations

Limit representations

Generating functions

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

Related transformations

Involving KelvinBei[ν,z]

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Brychkov Yu.A. (2005)

Brychkov Yu.A. (2005)

Functional identities

Relations between contiguous functions

Differentiation

Low-order differentiation

With respect to ν

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

Definite integration

Integral transforms

Laplace transforms

Mellin transforms

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQRegularized

Involving HypergeometricPFQ

Through Meijer G

Classical cases for the direct function itself

Classical cases for powers of  KelvinBer

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Classical cases for products of  KelvinBer

Classical cases involving powers of KelvinBei

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Classical cases involving KelvinBei

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Classical cases involving KelvinKei

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Classical cases involving KelvinKer

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Classical cases involving KelvinBei, KelvinKer and KelvinKei

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Classical cases involving Bessel BesselJ

Classical cases involving Bessel BesselI

Classical cases involving Bessel BesselK

Classical cases involving Hypergeometric0F1

Classical cases involving Hypergeometric0F1Regularized

Generalized cases for the direct function itself

Generalized cases for powers of  KelvinBer

Brychkov Yu.A. (2006)

Generalized cases for products of  KelvinBer

Generalized cases involving powers of KelvinBei

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Generalized cases involving KelvinBei

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Generalized cases involving KelvinKei

Brychkov Yu.A. (2006)

Generalized cases involving KelvinKer

Brychkov Yu.A. (2006)

Generalized cases involving KelvinBei, KelvinKer and KelvinKei

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Generalized cases involving Bessel BesselJ

Generalized cases involving Bessel BesselI

Generalized cases involving Bessel BesselK

Generalized cases involving Hypergeometric0F1

Generalized cases involving Hypergeometric0F1Regularized

Through other functions

Representations through equivalent functions

With related functions

Theorems

History

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