EllipticE

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

E(z) is an analytical function of z which is defined over the whole complex z-plane.

Symmetries and periodicities

Mirror symmetry

Periodicity

The function E(z) is not periodic.

Poles and essential singularities

The function E(z) does not have poles and essential singularities.

Branch points

Branch cuts

Series representations

Generalized power series

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

Expansions on branch cuts

Expansions at z==0

Expansions at z==1

Expansions at z==∞

Residue 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

Identities

Functional identities

Complex characteristics

Real part

Imaginary part

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Definite integration

For the direct function itself

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric2F1

Through Meijer G

Classical cases for the direct function itself

Classical cases involving algebraic functions

Classical cases involving unit step UnitStep

Generalized cases involving algebraic functions

Generalized cases involving unit step UnitStep

Through other functions

Involving incomplete elliptic integrals

Involving Legendre functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

Theorems

The perimeter of an ellipse

The perimeter s of an ellipse with semi-axes a and b is s=4 b E(1-((a^2)/(b^2))).

The gravitational or electrostatic potential of a uniform circular ring

The gravitational or electrostatic potential V(r) of a uniform circular ring of radius (Subscript[r, 0]) is given by

V(r)∝(∫_0^(2 π))((cos(ϕ))/(Sqrt[(r^2)+((Subscript[r, 0])^2)+(z^2)-2 r (Subscript[r, 0]) cos(ϕ)]))dϕ==
	(2/(r (Subscript[r, 0]) (Sqrt[(z^2)+(((r+(Subscript[r, 0])))^2)]))) (((r^2)+(z^2)+(Subscript[r, 0]^2))K(-((4 r (Subscript[r, 0]))/((z^2)+(((r+(Subscript[r, 0])))^2)))) -((z^2)+(((r+(Subscript[r, 0])))^2))E(-((4 r (Subscript[r, 0]))/((z^2)+(((r+(Subscript[r, 0])))^2)))) )

History

– J. Wallis (1655)
– A. M. Legendre (1811)
– C. G. J. Jacobi (1829)

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