elliptic integrals and Jacobi elliptic functions

For a modulusMathworldPlanetmathPlanetmathPlanetmath 0<k<1 (while here, we define the complementary modulus to k to be the positive number k with k2+k2=1) , write

F(ϕ,k) = 0ϕdθ1-k2sin2θ (1)
E(ϕ,k) = 0ϕ1-k2sin2θ𝑑θ (2)
Π(n,ϕ,k) = 0ϕdθ(1+nsin2θ)1-k2sin2θ (3)

The change of variable x=sinϕ turns these into

F1(x,k) = 0xdv(1-v2)(1-k2v2) (4)
E1(x,k) = 0x1-k2v21-v2𝑑v (5)
Π1(n,x,k) = 0xdv(1+nv2)(1-v2)(1-k2v2) (6)

The first three functionsMathworldPlanetmath are known as Legendre’s form of the incomplete elliptic integrals of the first, second, and third kinds respectively. Notice that (2) is the special case n=0 of (3). The latter three are known as Jacobi’s form of those integralsDlmfPlanetmath. If ϕ=π/2, or x=1, they are called completePlanetmathPlanetmathPlanetmath rather than incomplete integrals, and we refer to the auxiliary elliptic integrals K(k)=F(π/2,k), E(k)=E(π/2,k), etc.

One use for elliptic integrals is to systematize the evaluation of certain other integrals. In particular, let p be a third- or fourth-degree polynomialPlanetmathPlanetmath in one variable, and let y=p(x). If q and r are any two polynomials in two variables, then the indefinite integral


has a “closed formMathworldPlanetmathPlanetmath” in terms of the above incomplete elliptic integrals, together with elementary functionsMathworldPlanetmath and their inverses.

Jacobi’s elliptic functionsMathworldPlanetmath

In (1) we may regard ϕ as a function of F, or vice versa. The notation used is

ϕ=amu  u=argϕ

and ϕ and u are known as the amplitude and argumentMathworldPlanetmath respectively. But x=sinϕ=sinamu. The function usinamu=x is denoted by sn and is one of four Jacobi (or JacobianDlmfPlanetmath) elliptic functions. The four are:

snu = x
cnu = 1-x2
tnu = snucnu
dnu = 1-k2x2

When the Jacobian elliptic functionsDlmfDlmfDlmfDlmfDlmfDlmfDlmf are extended to complex arguments, they are doubly periodic and have two poles in any parallelogram of periods; both poles are simple.

Title elliptic integrals and Jacobi elliptic functions
Canonical name EllipticIntegralsAndJacobiEllipticFunctions
Date of creation 2013-03-22 13:58:28
Last modified on 2013-03-22 13:58:28
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Definition
Classification msc 33E05
Related topic ArithmeticGeometricMean
Related topic PerimeterOfEllipse
Defines elliptic integral
Defines Jacobi elliptic function
Defines Jacobian elliptic function
Defines complementary modulus
Defines complete elliptic integral