Let $s^2(t)$ be a cubic or quartic polynomial in $t$ with simple zeros, and let $r(s,t)$ be a rational function of $s$ and $t$ containing at least one odd power of $s$. Then

is called an elliptic integral. Because $s^2$ is a polynomial, we have

where $p_j$ is a polynomial in $t$ while $\rho$ and $\sigma$ are rational functions of $t$. Thus the elliptic part of (19.2.1) is

Assume $1-{\sin }^2\varphi \in ℂ\setminus (-\mathrm{\infty },0]$ and $1-k^2{\sin }^2\varphi \in ℂ\setminus (-\mathrm{\infty },0]$, except that one of them may be 0, and $1-{\alpha }^2{\sin }^2\varphi \in ℂ\setminus \{0\}$. Then

The paths of integration are the line segments connecting the limits of integration. The integral for $E(\varphi ,k)$ is well defined if $k^2={\sin }^2\varphi =1$, and the Cauchy principal value (§1.4(v)) of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ is taken if $1-{\alpha }^2{\sin }^2\varphi$ vanishes at an interior point of the integration path. Also, if $k^2$ and ${\alpha }^2$ are real, then $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ is called a circular or hyperbolic case according as ${\alpha }^2({\alpha }^2-k^2)({\alpha }^2-1)$ is negative or positive. The circular and hyperbolic cases alternate in the four intervals of the real line separated by the points ${\alpha }^2=0,k^2,1$.

The cases with $\varphi =\pi /2$ are the complete integrals:

The principal branch of $K\left(k\right)$ and $E\left(k\right)$ is $|\mathrm{ph}\left(1-k^2\right)|\le \pi$, that is, the branch-cuts are $(-\mathrm{\infty },-1]\cup [1,+\mathrm{\infty })$. The principal values of $K\left(k\right)$ and $E\left(k\right)$ are even functions.

Legendre’s complementary complete elliptic integrals are defined via

with a branch point at $k=0$ and principal branch $|\mathrm{ph}k|\le \pi$. Let $k^{\prime }=\sqrt{1-k^2}$. Then

For more details on the analytical continuation of these complete elliptic integrals see Lawden (1989, §§8.12–8.14).

If $m$ 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 $a,b,p$ are real parameters, and $k_c$ and $x$ are real or complex variables, with $p\ne 0$, $k_c\ne 0$. If , then the integral in (19.2.11) is a Cauchy principal value.

With

special cases include

and

The integrals are complete if $x=\mathrm{\infty }$. If , then $k_c$ is pure imaginary.

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

Let $x\in ℂ\setminus (-\mathrm{\infty },0)$ and $y\in ℂ\setminus \{0\}$. We define

where the Cauchy principal value is taken if . Formulas involving $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using $R_C(x,y)$.

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 $x$ and $y$ are positive, $R_C(x,y)$ is an inverse circular function if and an inverse hyperbolic function (or logarithm) if $x>y$:

The Cauchy principal value is hyperbolic:

For the special cases of $R_C(x,x)$ and $R_C(0,y)$ see (19.6.15).

If the line segment with endpoints $x$ and $y$ lies in $ℂ\setminus (-\mathrm{\infty },0]$, then