The bestknown 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:
