HarmonicNumber2

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed z

For fixed r

General characteristics

Domain and analyticity

(Subscript[H, z]^((r))) is an analytical function of z and r which is defined in (ℂ^2). For fixed z, it is an entire function of r.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to r

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

Branch points

With respect to r

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

Branch cuts

With respect to r

For fixed z, the function (Subscript[H, z]^((r))) does not have branch cuts.

Series representations

Generalized power series

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

Expansions at z==0

Expansions at z==-m

Expansions at r==1

Asymptotic series expansions

Residue representations

Other series representations

Integral representations

On the real axis

Of the direct function

Contour integral representations

Generating functions

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Multiple arguments

Products, sums, and powers of the direct function

Sums of the direct function

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations of special kind

Differentiation

Low-order differentiation

With respect to z

With respect to r

Symbolic differentiation

With respect to z

With respect to r

Fractional integro-differentiation

With respect to z

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Involving one direct function and with respect to r

Integral transforms

Laplace transforms

Summation

Finite summation

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQ

Through Meijer G

Classical cases for the direct function itself

Representations through equivalent functions

With related functions

History

– L. Euler (1740)

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