Let be a cubic or quartic polynomial in with simple zeros, and let be a rational function of and containing at least one odd power of . Then
is called an elliptic integral. Because is a polynomial, we have
where is a polynomial in while and are rational functions of . Thus the elliptic part of (19.2.1) is
Assume and , except that one of them may be 0, and . Then
The paths of integration are the line segments connecting the limits of integration. The integral for is well defined if , and the Cauchy principal value (§1.4(v)) of is taken if vanishes at an interior point of the integration path. Also, if and are real, then is called a circular or hyperbolic case according as is negative or positive. The circular and hyperbolic cases alternate in the four intervals of the real line separated by the points .
The cases with are the complete integrals:
The principal branch of and is , that is, the branch-cuts are . The principal values of and are even functions.
Legendre’s complementary complete elliptic integrals are defined via
with a branch point at and principal branch . Let . Then
For more details on the analytical continuation of these complete elliptic integrals see Lawden (1989, §§8.12–8.14).
If is an integer, then
Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). Three are defined by
Here are real parameters, and and are real or complex variables, with , . If , then the integral in (19.2.11) is a Cauchy principal value.
special cases include
The integrals are complete if . If , then is pure imaginary.
Lastly, corresponding to Legendre’s incomplete integral of the third kind we have
Let and . We define
where the Cauchy principal value is taken if . Formulas involving that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using .
In (19.2.18)–(19.2.22) the inverse trigonometric and hyperbolic functions assume their principal values (§§4.23(ii) and 4.37(ii)). When and are positive, is an inverse circular function if and an inverse hyperbolic function (or logarithm) if :
The Cauchy principal value is hyperbolic:
For the special cases of and see (19.6.15).
If the line segment with endpoints and lies in , then