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单词 EllipticIntegralsAndJacobiEllipticFunctions
释义

elliptic integrals and Jacobi elliptic functions


Elliptic integralsMathworldPlanetmath

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 incompleteelliptic 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 incompleteintegrals, 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 ofcertain other integrals.In particular, let p be a third- or fourth-degree polynomialPlanetmathPlanetmathin one variable, and let y=p(x).If q and r are any two polynomials in two variables, then theindefinite integral

q(x,y)r(x,y)𝑑x

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=xis 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 ofperiods; both poles are simple.

Titleelliptic integrals and Jacobi elliptic functions
Canonical nameEllipticIntegralsAndJacobiEllipticFunctions
Date of creation2013-03-22 13:58:28
Last modified on2013-03-22 13:58:28
Ownermathcam (2727)
Last modified bymathcam (2727)
Numerical id7
Authormathcam (2727)
Entry typeDefinition
Classificationmsc 33E05
Related topicArithmeticGeometricMean
Related topicPerimeterOfEllipse
Defineselliptic integral
DefinesJacobi elliptic function
DefinesJacobian elliptic function
Definescomplementary modulus
Definescomplete elliptic integral
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