# 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.

Common argument conventions for elliptic integrals and elliptic functions.

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

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 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 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

.

Out[1]= | |

Here is

with

.

Out[2]= | |

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

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,{g_{2},g_{3}}] | Weierstrass elliptic function |

WeierstrassPPrime[u,{g_{2},g_{3}}] | derivative of Weierstrass elliptic function |

InverseWeierstrassP[p,{g_{2},g_{3}}] | inverse Weierstrass elliptic function |

WeierstrassSigma[u,{g_{2},g_{3}}] | Weierstrass sigma function |

WeierstrassZeta[u,{g_{2},g_{3}}] | 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

.

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

by

. 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, {g_{2}, g_{3}}].

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

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

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 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 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.