§19.1 Special Notation

Keywords:: Bulirsch’s elliptic integrals, Legendre’s elliptic integrals, elliptic integrals, symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.1

(For other notation see Notation for the Special Functions.)

$l,m,n$	nonnegative integers.
$\phi$	real or complex argument (or amplitude).
$k$	real or complex modulus.
$k^{{\prime}}$	complementary real or complex modulus, $k^{2}+{k^{{\prime}}}^{2}=1$ .
$\alpha^{2}$	real or complex parameter.
$\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$	beta function (§5.12).

All square roots have their principal values. All derivatives are denoted by differentials, not by primes.

The first set of main functions treated in this chapter are Legendre’s complete integrals

19.1.1

$\mathop{K\/}\nolimits\!\left(k\right),$

$\mathop{E\/}\nolimits\!\left(k\right),$

$\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right),$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : Legendre’s complete elliptic integral of the third kind, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter
Referenced by:: §19.1, §19.1
Permalink:: http://dlmf.nist.gov/19.1.E1
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

of the first, second, and third kinds, respectively, and Legendre’s incomplete integrals

19.1.2

$\mathop{F\/}\nolimits\!\left(\phi,k\right),$

$\mathop{E\/}\nolimits\!\left(\phi,k\right),$

$\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right),$

of the first, second, and third kinds, respectively. This notation follows Byrd and Friedman (1971, 110). We use also the function $\mathop{D\/}\nolimits\!\left(\phi,k\right)$ , introduced by Jahnke et al. (1966, p. 43). The functions (19.1.1) and (19.1.2) are used in Erdélyi et al. (1953b, Chapter 13), except that $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ and $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ are denoted by $\Pi _{1}(\nu,k)$ and $\Pi(\phi,\nu,k)$ , respectively, where $\nu=-\alpha^{2}$ .

In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by $K(\alpha)$ , $E(\alpha)$ , $\Pi(n\backslash\alpha)$ , $F(\phi\backslash\alpha)$ , $E(\phi\backslash\alpha)$ , and $\Pi(n;\phi\backslash\alpha)$ , where $\alpha=\mathop{\mathrm{arcsin}\/}\nolimits k$ and $n$ is the $\alpha^{2}$ (not related to $k$ ) in (19.1.1) and (19.1.2). Also, frequently in this reference $\alpha$ is replaced by $m$ and $\mathord{\backslash}\alpha$ by $\mathord{|}m$ , where $m=k^{2}$ . However, it should be noted that in Chapter 8 of Abramowitz and Stegun (1964) the notation used for elliptic integrals differs from Chapter 17 and is consistent with that used in the present chapter and the rest of the NIST Handbook and DLMF.

The second set of main functions treated in this chapter is

19.1.3

$\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ ,

$\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ ,

$\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ ,

$\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ ,

$\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ ,

$\mathop{R_{{-a}}\/}\nolimits\!\left(b_{1},b_{2},\dots,b_{n};z_{1},z_{2},\dots,z_{n}\right).$

$\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ , $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ , and $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ are the symmetric (in $x$ , $y$ , and $z$ ) integrals of the first, second, and third kinds; they are complete if exactly one of $x$ , $y$ , and $z$ is identically 0.

$\mathop{R_{{-a}}\/}\nolimits\!\left(b_{1},b_{2},\dots,b_{n};z_{1},z_{2},\dots,z_{n}\right)$ is a multivariate hypergeometric function that includes all the functions in (19.1.3).

A third set of functions, introduced by Bulirsch (1965b, a, 1969a), is

19.1.4

$\mathop{\mathrm{el1}\/}\nolimits\!\left(x,k_{c}\right),$

$\mathop{\mathrm{el2}\/}\nolimits\!\left(x,k_{c},a,b\right),$

$\mathop{\mathrm{el3}\/}\nolimits\!\left(x,k_{c},p\right),$

$\mathop{\mathrm{cel}\/}\nolimits\!\left(k_{c},p,a,b\right).$

Symbols:: $\mathop{\mathrm{cel}\/}\nolimits\!\left(k_{c},p,a,b\right)$ : Bulirsch’s complete elliptic integral, $\mathop{\mathrm{el1}\/}\nolimits\!\left(x,k_{c}\right)$ : Bulirsch’s incomplete elliptic integral of the first kind, $\mathop{\mathrm{el2}\/}\nolimits\!\left(x,k_{c},a,b\right)$ : Bulirsch’s incomplete elliptic integral of the second kind, $\mathop{\mathrm{el3}\/}\nolimits\!\left(x,k_{c},p\right)$ : Bulirsch’s incomplete elliptic integral of the third kind and $k$ : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.1.E4
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

The first three functions are incomplete integrals of the first, second, and third kinds, and the $\mathop{\mathrm{cel}\/}\nolimits$ function includes complete integrals of all three kinds.