§19.2 Definitions

§19.2(i) General Elliptic Integrals

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

19.2.1

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

§19.2(ii) Legendre’s Integrals

Assume and , except that one of them may be 0, and . Then

19.2.4
19.2.5
19.2.7

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:

19.2.8
19.2.9

If is an integer, then

§19.2(iii) Bulirsch’s Integrals

Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). Two are defined by

19.2.11
19.2.12

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.

Lastly, corresponding to Legendre’s incomplete integral of the third kind we have

§19.2(iv) A Related Function:

Let and . We define

19.2.17

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: