EllipticPi

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed n

For fixed m

Values at infinities

General characteristics

Domain and analyticity

Π(n|m) is an analytical function of n and m which is defined over (ℂ^2).

Symmetries and periodicities

Mirror symmetry

Periodicity

The function Π(n|m) is not periodic.

Poles and essential singularities

With respect to m

The function Π(n|m) does not have poles and essential singularities with respect to m.

With respect to n

The function Π(n|m) does not have poles and essential singularities with respect to n.

Branch points

With respect to m

With respect to n

Branch cuts

With respect to n

With respect to m

Series representations

Generalized power series

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

Expansions on branch cuts

Expansions at n==0

Expansions at n==1

Expansions at n==∞

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

Expansions on branch cuts

Expansions at m==0

Expansions at m==1

Expansions at m==∞

Expansions at {m,n}=={0,0}

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

With respect to n

With respect to m

Identities

Functional identities

Differentiation

Low-order differentiation

With respect to n

With respect to m

Symbolic differentiation

With respect to n

With respect to m

Fractional integro-differentiation

With respect to n

With respect to m

Integration

Indefinite integration

Involving only one direct function with respect to n

Involving only one direct function with respect to m

Involving one direct function and elementary functions with respect to m

Representations through more general functions

Through hypergeometric functions of two variables

Through Meijer G

Classical cases

Through other functions

Involving incomplete elliptic integrals

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

History

– A. M. Legendre (1811)

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