Introductions to EllipticPi3

Introduction to the incomplete elliptic integrals

General

Elliptic integrals were encountered in the work of J. Wallis (1655–1659) who investigated the integral in modern notation:

Broader interest in such integrals was stimulated by the appearance and development of integral calculus in the 18th and 19th centuries. Mathematicians found that integrals, containing quadratic polynomials under a square root of the form:

can be evaluated through elementary functions by formulas such as:

But these early mathematicians could not find simple formulas for similar integrals containing higher-degree polynomials:

Many important applications of these integrals were found at that time. The problem of evaluating such integrals was converted into the problem of evaluating only three basic integrals. They were later denoted by their special notation and named the incomplete elliptic integrals of the first, second, and third kinds—F(z|m), E(z|m), and Π(n;z|m) (A. M. Legendre):

The corresponding definite integrals (for z==(π/2)) were named the complete elliptic integrals of the first, second, and third kinds denoted by the symbols K(z), E(z), and Π(n|m):

These integrals were extensively studied for another important reason—development of the theory of the double periodic functions. These functions were called elliptic functions. The elliptic integrals and elliptic functions were studied simultaneously on several occasions throughout history and a deep connection exists between these two areas of mathematics. The following chronology reflects the main steps in building the theory of elliptic integrals.

L. Euler (1733, 1757, 1763, 1766) derived the addition theorem for the incomplete elliptic integrals F(z|m), and E(z|m).

J.-L. Lagrange (1783) and especially A. M. Legendre (1793, 1811, 1825–1828) devoted a lot of attention to the study of the different properties of those two elliptic integrals. C. F. Gauss (1799, 1818) also used these integrals during his research.

Simultaneously, A. M. Legendre (1811) introduced the incomplete elliptic integral of the third kind and the complete versions of all three elliptic integrals.

C. G. J. Jacobi (1827–1829)  introduced inverse functons of the elliptic integrals F(z|m) and E(z|m), which led him to build the theory of elliptic functions. In 1829, C. G. J. Jacobi defined the following function:

which was later called the Jacobi zeta function. J. Liouville (1840) also studied elliptic integrals F(z|m) and E(z|m).

N. H. Abel, independently from C. G. J. Jacobi, got some of his results and studied the so-called hyperelliptic and Abelian integrals.

Definitions of incomplete elliptic integrals

The incomplete elliptic integral of the first kind F(z|m), incomplete elliptic integral of the second kind E(z|m), incomplete elliptic integral of the third kind Π(n;z|m), and Jacobi zeta function Ζ(z|m) are defined by the following formulas:

The previous functions are called incomplete elliptic integrals.

A quick look at the incomplete elliptic integrals

Here is a quick look at the graphics for the incomplete elliptic integrals along the real axis.

Connections within the group of incomplete elliptic integrals and with other function groups

Representations through more general functions

Through elliptic theta functions, for example:

Through inverse Jacobi elliptic functions, for example:

Through Weierstrass elliptic functions and inverse elliptic nome (q^(-1))(t), for example:

Representations through related functions

Relations to inverse functions

Representations through other incomplete elliptic integrals

The best-known properties and formulas for incomplete elliptic integrals

Simple values at zero

Specific values

Analyticity

The incomplete elliptic integrals F(z|m), E(z|m), and Ζ(z|m) are analytical functions of z and m, which are defined over (ℂ^2).

The incomplete elliptic integral Π(n;z|m) is an analytical function of n, z, and m, which is defined over (ℂ^3).

Poles and essential singularities

Branch points and branch cuts

Periodicity

Parity and symmetry

Series representations

Other series representations

Integral representations

Transformations

Identities

Representations of derivatives

Integration

Differential equations

Applications of incomplete elliptic integrals

Applications of incomplete elliptic integrals include geometry, physics, mechanics, electrodynamics, statistical mechanics, astronomy, geodesy, geodesics on conics, and magnetic field calculations.

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