StruveH

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed ν

For fixed z

Values at fixed points

General characteristics

Domain and analyticity

(Subscript[H, 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[H, 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[H, ν])(z) has only one singular point at ν =(OverTilde[∞]).  It is an essential singular point.

Branch points

With respect to z

For fixed noninteger ν, the function (Subscript[H, ν])(z) has two branch points: z==0, z==(OverTilde[∞]). At the same time, the point z==(OverTilde[∞]) is an essential singularity.

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

With respect to ν

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

Branch cuts

With respect to z

When ν is an integer, (Subscript[H, ν])(z) is an entire function of z. For fixed noninteger ν, it has one infinitely long branch cut.  For fixed noninteger ν, the function  (Subscript[H, ν])(z) is a single-valued function on the z-plane cut along the interval (-∞, 0), where it is continuous from above.

With respect to ν

For fixed z, the function (Subscript[H, 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 on branch cuts

Expansions at z==0

Asymptotic series expansions

Expansions inside Stokes sectors

The general formulas

Expansions for any z  in exponential form

Expansions for any z  in trigonometric form

Residue representations

Other series representations

Integral representations

On the real axis

Of the direct function

Contour integral representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Brychkov Yu.A. (2005)

Brychkov Yu.A. (2005)

Brychkov Yu.A. (2005)

Brychkov Yu.A. (2005)

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)

Brychkov Yu.A. (2005)

Brychkov Yu.A. (2005)

Brychkov Yu.A. (2005)

With respect to z

Symbolic differentiation

With respect to z

Brychkov Yu.A. (2005)

Brychkov Yu.A. (2005)

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 and elementary functions

Involving direct function and Bessel-, Airy-, Struve-type 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

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQRegularized

Involving HypergeometricPFQ

Through Meijer G

Classical cases for the direct function itself

Classical cases involving Bessel BesselY

Generalized cases for the direct function itself

Generalized cases involving Bessel BesselY

Through other functions

Representations through equivalent functions

With related functions

Inequalities

Theorems

Struve H-Transformation

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

History

– H. Struve (1882)

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