The best-known properties and formulas for complete elliptic integrals

Real values for real restricted arguments

For real values of arguments , , and (with , , ) the values of all complete elliptic integrals , , and are real.

Simple values at zero

All complete elliptic integrals , , and are equal to at the origin:

Specific values

All complete elliptic integrals , , and can be represented through elementary or other functions when , , or , or , or or :

At any infinity, the complete elliptic integrals , , and have the following values:

Analyticity

The complete elliptic integrals and are analytical functions of , which are defined over the whole complex ‐plane.

The complete elliptic integral is an analytical function of and , which is defined over .

Poles and essential singularities

All complete elliptic integrals , , and do not have poles and essential singularities.

Branch points and branch cuts

The complete elliptic integrals and have two branch points: and .

They are single‐valued functions on the ‐plane cut along the interval . They are continuous from below on the interval :

For fixed , the function has two branch points at and . For fixed , the function has two branch points at and .

Periodicity

All complete elliptic integrals , , and are not periodical functions.

Parity and symmetry

All complete elliptic integrals , , and have mirror symmetry:

Series representations

All complete elliptic integrals , , and have the following series expansions at the point :

The complete elliptic integrals and have the following series expansions at the point :

The complete elliptic integrals and have the following series expansions at the point :

The complete elliptic integral has the following series expansions at the point :

The complete elliptic integral has the following series expansions at the point :

The complete elliptic integral has the following series expansions at the point :

The complete elliptic integral has the following series expansions at the point :

The previous formulas can be rewritten in summed forms of the truncated series expansion near corresponding points , , or :

Other series representations

Some elliptic integrals have special series representations through the elliptic nome and inverse Jacobi elliptic functions by the formulas:

Integral representations

The complete elliptic integrals , , and have the following integral representations:

Identities

The complete elliptic integrals , , and satisfy numerous identities, for example:

Representations of derivatives

The first derivatives of all complete elliptic integrals , , and with respect to their variables can also be represented through complete elliptic integrals by the following formulas:

The symbolic derivatives of all complete elliptic integrals , , and with respect to their variables can be represented through Gauss classical or regularized hypergeometric functions by the following formulas:

Integration

The indefinite integrals of all complete elliptic integrals , , and with respect to their variables can be expressed through complete elliptic integrals (or through hypergeometric functions of two variables) by the following formulas:

Differential equations

All complete elliptic integrals , , and satisfy ordinary linear differential equations: