The bestknown 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 arbitraryorder 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 arbitraryorder 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:
