EllipticK

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

Values at infinities

Singular values

General characteristics

Domain and analyticity

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

Symmetries and periodicities

Mirror symmetry

Periodicity

The function K(z) is not periodic.

Poles and essential singularities

The function K(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

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

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

Classical cases involving  Sign

Classical cases involving powers of complete elliptic integral EllipticK

Generalized cases involving algebraic functions

Generalized cases involving unit step UnitStep

Generalized cases involving  Sign

Generalized cases involving powers of complete elliptic integral EllipticK

Through other functions

Involving incomplete elliptic integrals

Involving elliptic theta functions

Involving inverse Jacobi functions

Involving some elliptic-type functions

Involving Legendre functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With inverse function

With related functions

Theorems

The period T of a mathematical pendulum in a gravitational field

The period T of a mathematical pendulum of length l in a gravitational field with acceleration g and maximal angle of excursion  α is given by T==4(Sqrt[(l/g)])K((sin^2)((α/2))).

The partition function for a one-dimensional monatomic ideal classical gas

The partition function Z for a one-dimensional monatomic ideal classical gas of n atoms in a box of length l at temperature T is given by  Z==(1/((2^n)n!))((((Sqrt[(2/π)K((q^(-1))(exp(-((3((λ(T))^2))/(8π (l^2))))))])-1))^n), where λ(T) is the thermal de Broglie wavelength.

The magnetic induction of an infiniely long selenoid

The magnetic induction B of an infinitely long solenoid formed by a wire (parametrized by ϕ) {R cos(ϕ),R sin(ϕ),R ϕ tan(α)} carrying the current (Subscript[i, 0]) is at the center line is given by

B∝{0,-(Subscript[i, 0])(cot(α) (Subscript[K, 0])(cot(α))+(Subscript[K, 1])(cot(α))),(Subscript[i, 0])((cot(α))/(2π R))}.

The lattice Green function for the body-centered cubic lattice

The lattice Green function G(ε)==(1/(π^3))(∫_0^π)(∫_0^π)(∫_0^π)(((1-(ε/3)(cos(x)cos(y)cos(z))))^(-1))dxdydz for the simple cubic lattice can be expressed as

(4/(π^2))(Sqrt[1-(3/4)(α^2)])(1/(1-α))K((1/2)+(β/4)(Sqrt[4-β])-(((2-β))/4)(Sqrt[1-β]))K((1/2)-(β/4)(Sqrt[4-β])-(((2-β))/4)(Sqrt[1-β]))/;α==(1/2)+((ε^2)/6)-(1/2)(Sqrt[1-(ε^2)])(Sqrt[1-((ε^2)/9)])∧β==(α/(α-1)).

The probability that a random walk in three dimensions will return to its origin

The probability (Subscript[p, 0]) that a random walk in three dimensions will return to its point of origin is given by

(Subscript[p, 0])=1-((π^2)/72)(6+2 (Sqrt[3])+(Sqrt[6])) ((K(35+24 (Sqrt[2])-20 (Sqrt[3])-14 (Sqrt[6])))^(-2)) ≈0.34053732955099914283….

History

– A. M. Legendre (1811, 1825)
– C. G. J. Jacobi (1829)

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