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Elliptic Integrals and Elliptic Functions
| Elliptic Integrals | Elliptic Modular Functions |
| Elliptic Functions | Generalized Elliptic Integrals and 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=60sum_(r, s)^(')w^(-4), g_3=140sum_(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](Files/EllipticIntegralsAndEllipticFunctions.en/Image_1.png "◼ 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=60sum_(r, s)^(')w^(-4), g_3=140sum_(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  |
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[, m] is given for 
by 
. 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[, m] is given for 
by 
.
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[, m] is given by 
.
The Heuman lambda function is given by 
.
The elliptic integral of the third kind EllipticPi[n, , m] is given by 
.
The complete elliptic integral of the third kind EllipticPi[n, m] is given by 
.
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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[u, m] 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[u, m] and JacobiCN[u, m] are given respectively by 
and 
, where 
. In addition, JacobiDN[u, m] 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 
.
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Also built into Mathematica are the inverse Jacobi elliptic functions InverseJacobiSN[v, m], InverseJacobiCN[v, m], 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[a, u, q] 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 
by 
. In addition, 
is sometimes replaced by 
, given by 
. All the theta functions satisfy a diffusion-like differential equation 
.
The Siegel theta function SiegelTheta[, s] 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[
, , s] 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[u, {g2, g3}] can be considered as the inverse of an elliptic integral. The Weierstrass function 
gives the value of 
for which 
. The function WeierstrassPPrime[u, {g2, g3}] 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[p, {g2, g3}] 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[{p, q}, {g2, g3}] 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[u, {g2, g3}] and Weierstrass sigma function WeierstrassSigma[u, {g2, g3}] 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  |
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[{x, y}, {a, b}] 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[u, {a, b}] is the inverse of EllipticLog. It returns the list 
that appears in EllipticLog. EllipticExp is an elliptic function, doubly periodic in the complex 
plane.
ArithmeticGeometricMean[a, b] gives the arithmetic-geometric mean (AGM) of two numbers 
and 
. This quantity is central to many numerical algorithms for computing elliptic integrals and other functions. For positive reals 
and 
the AGM is obtained by starting with 
, 
, then iterating the transformation 
, 
until 
to the precision required.
