KelvinBei

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

bei(z) is an entire, and so analytic, function of z, which is defined in the whole complex z-plane.

Symmetries and periodicities

Parity

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

The function bei(z) has only one singular point at z = (OverTilde[∞]).  It is an essential singular point.

Branch points

The function bei(z) does not have branch points.

Branch cuts

The function bei(z) does not have branch cuts.

Series representations

Generalized power series

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

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

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 KelvinBer[z]

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

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  KelvinBei

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Classical cases involving powers of KelvinBer

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Classical cases involving KelvinBer

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 KelvinBer, 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

Generalized cases for the direct function itself

Generalized cases for powers of  KelvinBei

Brychkov Yu.A. (2006)

Generalized cases involving powers of KelvinBer

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Generalized cases involving KelvinBer

Brychkov Yu.A. (2006)

Generalized cases involving KelvinKei

Brychkov Yu.A. (2006)

Generalized cases involving KelvinKer

Brychkov Yu.A. (2006)

Generalized cases involving KelvinBer, 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

Representations through equivalent functions

With related functions

Theorems

History

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