The best-known properties and formulas for incomplete elliptic integrals

Simple values at zero

The incomplete elliptic integrals , , and are equal to at the origin points:

Specific values

The incomplete elliptic integrals , , and for particular argument values can be evaluated in closed forms, for example:

At infinities, the incomplete elliptic integrals , , and have the following values:

Analyticity

The incomplete elliptic integrals , and are analytical functions of and , which are defined over . The incomplete elliptic integral is an analytical function of , , and , which is defined over .

Poles and essential singularities

The incomplete elliptic integrals , , and do not have poles and essential singularities with respect to their variables.

Branch points and branch cuts

For fixed , the functions , , and have an infinite number of branch points at and . They have complicated branch cut.

For fixed , , the function has two branch points at and . For fixed , , the function has an infinite number of branch points at , and . For fixed , , the function does not have branch points.

Periodicity

The incomplete elliptic integrals , and are quasi‐periodic functions with respect to :

The incomplete elliptic integral is a periodic function with respect to with period :

Parity and symmetry

The incomplete elliptic integrals , , and have mirror symmetry:

The incomplete elliptic integrals , , and are odd functions with respect to .

Series representations

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

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

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

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

Other series representations

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

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

The incomplete elliptic integrals , , and can be represented through different kinds of series, for example:

where is an elliptic nome and is a complete elliptic integral.

Integral representations

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

Transformations

The incomplete elliptic integrals , , and with linear arguments can sometimes be simplified, for example:

In some cases, simplification can be realized for more complicated arguments, for example:

Sums of the incomplete elliptic integrals , , and with different values and can be evaluated by the following summation formulas:

Identities

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

Representations of derivatives

The first derivative of the incomplete elliptic integral with respect to variable has the following representation:

The previous formula can be generalized to the derivative:

The first derivatives of the incomplete elliptic integrals , , , and with respect to variable have the following simple representations:

The previous formulas can be generalized to the arbitrary-order symbolic derivatives:

The first derivatives of the incomplete elliptic integrals , , and with respect to variable have the following representations:

The previous formulas can be generalized to the arbitrary-order symbolic derivatives:

Integration

The indefinite integral of the incomplete elliptic integral with respect to variable has the following representation through the infinite series that includes the Appell hypergeometric function of two variables:

The indefinite integrals of all incomplete elliptic integrals , , , and with respect to variable have the following representations:

The indefinite integrals of incomplete elliptic integrals , , and with respect to variable can be expressed in closed forms by the following formulas:

Differential equations

The incomplete elliptic integrals , , , and satisfy the following differential equations: