LegendreQGeneral

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed ν

For fixed z

General characteristics

Domain and analyticity

(Subscript[Q, ν])(z) is an analytical function of ν and z which is defined over (ℂ^2).

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

For fixed ν, the function (Subscript[Q, ν])(z) does not have poles and essential singularities.

With respect to ν

For fixed z, the function (Subscript[Q, ν])(z) has an infinite set of singular points with respect to ν:
a) the points ν==- k/;k∈(SuperPlus[ℕ]), are the simple poles with residues (Subscript[P, k-1])(z);
b) ν== (OverTilde[∞]) is the point of convergence of poles, which is an essential singular point.

Branch points

With respect to z

With respect to ν

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

Branch cuts

With respect to z

With respect to ν

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

Series representations

Generalized power series

Expansions at z==0

Expansions at z==1

Expansions at z==-1

Expansions at z==∞

Integral representations

On the real axis

Of the direct function

Integral representations of negative integer order

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations between contiguous functions

Differentiation

Low-order differentiation

With respect to ν

With respect to z

Symbolic differentiation

With respect to ν

With respect to z

Fractional integro-differentiation

With respect to z

Summation

Infinite summation

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric2F1

Through Meijer G

Classical cases for the direct function itself

Through other functions

Involving some hypergeometric-type functions

Involving spheroidal functions

Representations through equivalent functions

With related functions

History

– D. Bernoulli (1748)
– A. M. Legendre (1782)
– E. Heine (1842)
– L. Schläfli (1881)

Copyright