Introductions to EllipticPi

Introduction to the complete elliptic integrals

General

Elliptic integrals were introduced in the investigations of J. Wallis (1655
–1659) who studied the integral (in modern notation):

L. Euler (1733, 1757, 1763, 1766) derived the addition theorem for the 
following elliptic integrals currently called incomplete elliptic integrals 
of the first and second kind:

J.-L. Lagrange (1783) and especially A. M. Legendre (1793, 1811, 1825
–1828) devoted considerable attention to study different properties of 
these 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 the integrals:

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

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

N. H. Abel independently derived some of C. G. J. Jacobi's results and 
studied the so-called hyperelliptic and Abelian integrals.

Definitions of complete elliptic integrals

The complete elliptic integral of the first kind K(z), the complete elliptic integral of the second kind E(z), and the complete elliptic integral of the third kind Π(n|m) are defined through the following formulas:

where the incomplete elliptic integrals of the first, second, and third kind 
are as follows:

A quick look at the complete elliptic integrals

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

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

Representations through more general functions

Through the elliptic theta functions:

Through the arithmetic geometric mean:

Through the Jacobi elliptic functions:

Through the Weierstrass elliptic functions and inverse elliptic nome (q^(-1))(t):

Through the Legendre P and Q functions:

Relations to inverse functions

Representations through other complete elliptic integrals

The best-known properties and formulas for complete elliptic 
integrals

Real values for real restricted arguments

For real values of arguments z, n, and m (with z<1, n<1, m<1) the values of all complete elliptic integrals K(z), E(z), and Π(n|m) are real.

Simple values at zero

All complete elliptic integrals K(z), E(z), and Π(n|m) are equal to (π/2) at the origin:

Specific values

Analyticity

The complete elliptic integrals K(z) and E(z) are analytical functions of z, which are defined over the whole complex z-plane.

The complete elliptic integral Π(n|m) is an analytical function of n and m, which is defined over (ℂ^2).

Poles and essential singularities

All complete elliptic integrals K(z), E(z), and Π(n|m) do not have poles and essential singularities.

Branch points and branch cuts

Periodicity

Parity and symmetry

Series representations

Other series representations

Integral representations

Identities

Representations of derivatives

Integration

Differential equations

Applications of complete elliptic integrals

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

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