MATHEMATICA TUTORIAL

Elliptic Integrals and Elliptic Functions

Even more so than for other special functions, you need to be very careful about the arguments you give to elliptic integrals and elliptic functions. There are several incompatible conventions in common use, and often these conventions are distinguished only by the specific names given to arguments or by the presence of separators other than commas between arguments.

$◼ Amplitude phi (used by Mathematica, in radians) ; ◼ Argument u (used by Mathematica): related to amplitude by phiam(u) ; ◼ Delta amplitude Delta(phi): Delta(phi)sqrt(1-msin^2(phi)) ; ◼ Coordinate x: xsin(phi) ; ◼ Characteristic n (used by Mathematica in elliptic integrals of the third kind) ; ◼ Parameter m (used by Mathematica): preceded by ❘, as in I(phi❘m) ; ◼ Complementary parameter m_1: m_11-m ; ◼ Modulus k: preceded by comma, as in I(phi,k); mk^2 ; ◼ Modular angle alpha: preceded by \ , as in I(phi\alpha); msin^2(alpha) ; ◼ Nome q: preceded by comma in theta functions; qexp[-pi K(1-m)/K{m)]exp(i pi omega^′/omega) ; ◼ Invariants g_2, g_3 (used by Mathematica) ; ◼ Half‐periods omega, omega^′: g_260∑_(r, s)^(′)w^(-4), g_3140∑_(r, s)^(′)w^(-6), where w2romega+2somega^′ ; ◼ Ratio of periods tau: tauomega^′/omega ; ◼ Discriminant Delta: Deltag_2^3-27g_3^2 ; ◼ Parameters of curve a, b (used by Mathematica) ; ◼ Coordinate y (used by Mathematica): related by y^2x^3+ax^2+b x$

Common argument conventions for elliptic integrals and elliptic functions.

JacobiAmplitude[u,m]	give the amplitude corresponding to argument u and parameter m
EllipticNomeQ[m]	give the nome q corresponding to parameter m
InverseEllipticNomeQ[q]	give the parameter m corresponding to nome q
WeierstrassInvariants[{,}]	give the invariants corresponding to the half-periods
WeierstrassHalfPeriods[{g₂,g₃}]	give the half-periods corresponding to the invariants

Converting between different argument conventions.

Elliptic Integrals

EllipticK[m]	complete elliptic integral of the first kind
EllipticF[,m]	elliptic integral of the first kind
EllipticE[m]	complete elliptic integral of the second kind
EllipticE[,m]	elliptic integral of the second kind
EllipticPi[n,m]	complete elliptic integral of the third kind
EllipticPi[n,,m]	elliptic integral of the third kind
JacobiZeta[,m]	Jacobi zeta function

Elliptic integrals.

Integrals of the form

, where

is a rational function, and

is a cubic or quartic polynomial in

, are known as elliptic integrals. Any elliptic integral can be expressed in terms of the three standard kinds of Legendre-Jacobi elliptic integrals.

The elliptic integral of the first kind EllipticF

is given for

. This elliptic integral arises in solving the equations of motion for a simple pendulum. It is sometimes known as an incomplete elliptic integral of the first kind.

Note that the arguments of the elliptic integrals are sometimes given in the opposite order from what is used in Mathematica.

The complete elliptic integral of the first kind EllipticK[m] is given by

. Note that

is used to denote the complete elliptic integral of the first kind, while

is used for its incomplete form. In many applications, the parameter

is not given explicitly, and

is denoted simply by

. The complementary complete elliptic integral of the first kind

is given by

. It is often denoted

and

give the "real" and "imaginary" quarter-periods of the corresponding Jacobi elliptic functions discussed in "Elliptic Functions".

The elliptic integral of the second kind EllipticE

is given for

The complete elliptic integral of the second kind EllipticE[m] is given by

. It is often denoted

. The complementary form is

The Jacobi zeta function JacobiZeta

is given by

The Heuman lambda function is given by

The elliptic integral of the third kind EllipticPi

is given by

The complete elliptic integral of the third kind EllipticPi

is given by

Here is a plot of the complete elliptic integral of the second kind

In[1]:=

Out[1]=

Here is

with

In[2]:=

Out[2]=

The elliptic integrals have a complicated structure in the complex plane.

In[83]:=

Out[1]=

Elliptic Functions

JacobiAmplitude[u,m]	amplitude function
JacobiSN[u,m], JacobiCN[u,m], etc.
	Jacobi elliptic functions , etc.
InverseJacobiSN[v,m], InverseJacobiCN[v,m], etc.
	inverse Jacobi elliptic functions , etc.
EllipticTheta[a,u,q]	theta functions (, ..., )
EllipticThetaPrime[a,u,q]	derivatives of theta functions (, ..., )
SiegelTheta[,s]	Siegel theta function
SiegelTheta[v,,s]	Siegel theta function
WeierstrassP[u,{g₂,g₃}]	Weierstrass elliptic function
WeierstrassPPrime[u,{g₂,g₃}]	derivative of Weierstrass elliptic function
InverseWeierstrassP[p,{g₂,g₃}]	inverse Weierstrass elliptic function
WeierstrassSigma[u,{g₂,g₃}]	Weierstrass sigma function
WeierstrassZeta[u,{g₂,g₃}]	Weierstrass zeta function

Elliptic and related functions.

Rational functions involving square roots of quadratic forms can be integrated in terms of inverse trigonometric functions. The trigonometric functions can thus be defined as inverses of the functions obtained from these integrals.

By analogy, elliptic functions are defined as inverses of the functions obtained from elliptic integrals.

The amplitude for Jacobi elliptic functions JacobiAmplitude

is the inverse of the elliptic integral of the first kind. If

, then

. In working with Jacobi elliptic functions, the argument

is often dropped, so

is written as

The Jacobi elliptic functions JacobiSN

and JacobiCN

are given respectively by

and

, where

. In addition, JacobiDN

is given by

There are a total of twelve Jacobi elliptic functions

, with the letters P and Q chosen from the set

, C, D and N. Each Jacobi elliptic function

satisfies the relation

, where for these purposes

There are many relations between the Jacobi elliptic functions, somewhat analogous to those between trigonometric functions. In limiting cases, in fact, the Jacobi elliptic functions reduce to trigonometric functions. So, for example,

and

The notation

is often used for the integrals

. These integrals can be expressed in terms of the Jacobi zeta function defined in "Elliptic Integrals".

One of the most important properties of elliptic functions is that they are doubly periodic in the complex values of their arguments. Ordinary trigonometric functions are singly periodic, in the sense that

for any integer

. The elliptic functions are doubly periodic, so that

for any pair of integers

and

The Jacobi elliptic functions

, etc. are doubly periodic in the complex

plane. Their periods include

and

, where

is the complete elliptic integral of the first kind.

The choice of "p" and "q" in the notation

for Jacobi elliptic functions can be understood in terms of the values of the functions at the quarter periods

and

This shows two complete periods in each direction of the absolute value of the Jacobi elliptic function

In[3]:=

Out[3]=

Also built into Mathematica are the inverse Jacobi elliptic functions InverseJacobiSN

, InverseJacobiCN

, etc. The inverse function

, for example, gives the value of

for which

. The inverse Jacobi elliptic functions are related to elliptic integrals.

The four theta functions

are obtained from EllipticTheta

by taking a to be

, or

. The functions are defined by

. The theta functions are often written as

with the parameter

not explicitly given. The theta functions are sometimes written in the form

, where

is related to

. In addition,

is sometimes replaced by

, given by

. All the theta functions satisfy a diffusion-like differential equation

The Siegel theta function SiegelTheta

with Riemann square modular matrix

of dimension p and vector s generalizes the elliptic theta functions to complex dimension p. It is defined by

, where n runs over all p-dimensional integer vectors. The Siegel theta function with characteristic SiegelTheta

is defined by

, where the characteristic

is a pair of p-dimensional vectors

The Jacobi elliptic functions can be expressed as ratios of the theta functions.

An alternative notation for theta functions is

, where

The Neville theta functions can be defined in terms of the theta functions as

, where

. The Jacobi elliptic functions can be represented as ratios of the Neville theta functions.

The Weierstrass elliptic function WeierstrassP

can be considered as the inverse of an elliptic integral. The Weierstrass function

gives the value of

for which

. The function WeierstrassPPrime

is given by

The Weierstrass functions are also sometimes written in terms of their fundamental half-periods

and

, obtained from the invariants

and

using WeierstrassHalfPeriods[{u, {g₂, g₃}].

The function InverseWeierstrassP

finds one of the two values of

for which

. This value always lies in the parallelogram defined by the complex number half-periods

and

InverseWeierstrassP

finds the unique value of

for which

and

. In order for any such value of

to exist,

and

must be related by

The Weierstrass zeta function WeierstrassZeta

and Weierstrass sigma function WeierstrassSigma

are related to the Weierstrass elliptic functions by

and

The Weierstrass zeta and sigma functions are not strictly elliptic functions since they are not periodic.

Elliptic Modular Functions

DedekindEta[]	Dedekind eta function
KleinInvariantJ[]	Klein invariant modular function
ModularLambda[]	modular lambda function

Elliptic modular functions.

The modular lambda function ModularLambda[

] relates the ratio of half-periods

to the parameter according to

The Klein invariant modular function KleinInvariantJ[

] and the Dedekind eta function DedekindEta[

] satisfy the relations

Modular elliptic functions are defined to be invariant under certain fractional linear transformations of their arguments. Thus for example

is invariant under any combination of the transformations

and

Generalized Elliptic Integrals and Functions

ArithmeticGeometricMean[a,b]	the arithmetic-geometric mean of and
EllipticExp[u,{a,b}]	generalized exponential associated with the elliptic curve
EllipticLog[{x,y},{a,b}]	generalized logarithm associated with the elliptic curve

Generalized elliptic integrals and functions.

The definitions for elliptic integrals and functions given above are based on traditional usage. For modern algebraic geometry, it is convenient to use slightly more general definitions.

The function EllipticLog

is defined as the value of the integral

, where the sign of the square root is specified by giving the value of

such that

. Integrals of the form

can be expressed in terms of the ordinary logarithm (and inverse trigonometric functions). You can think of EllipticLog as giving a generalization of this, where the polynomial under the square root is now of degree three.

The function EllipticExp