ParabolicCylinderD

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed ν

For fixed z

Brychkov Yu.A. (2005)

Brychkov Yu.A. (2005)

Brychkov Yu.A. (2005)

Brychkov Yu.A. (2005)

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

(Subscript[D, 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[D, ν])(z) has an essential singularity at z==(OverTilde[∞]).

With respect to ν

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

Branch points

With respect to z

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

With respect to ν

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

Branch cuts

With respect to z

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

With respect to ν

For fixed z, the function (Subscript[D, ν])(z) is an entire function of ν and 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

Integral representations

On the real axis

Of the direct function

Of the products of direct functions

Contour integral representations

Whittaker,Watson,16, Example12

Whittaker,Watson,16.6

Whittaker,Watson,16,Example 11

Integral representations of negative integer order

Rodrigues-type formula.

Limit representations

Generating functions

Differential equations

Ordinary linear differential equations and wronskians

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Addition formulas

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. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

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 one direct function and elementary functions

Definite integration

Integral transforms

Laplace transforms

Mellin transforms

Summation

Infinite summation

Operations

Limit operation

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQ

Involving Hypergeometric1F1Regularized

Involving hypergeometric HypergeometricU

Through Meijer G

Classical cases involving Exp

Classical cases involving Cosh

Classical cases involving Sinh

Classical cases for products of ParabolicCylinderD

Classical cases involving  Exp and Hermite HermiteH

Generalized cases involving Exp

Generalized cases involving Cosh

Generalized cases involving Sinh

Generalized cases for products of ParabolicCylinderD

Generalized cases involving  Exp and Hermite HermiteH

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

Theorems

History

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