§19.2 Definitions

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

19.2.1		$$\int r(s,t)dt$$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral and $r(s,t)$: rational function Referenced by: §19.2(i) Permalink: http://dlmf.nist.gov/19.2.E1 Encodings: TeX, pMML, png See also: Annotations for §19.2(i), §19.2 and Ch.19

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

19.2.2		$$r(s,t)=\frac{(p_1+p_{2}s)(p_3-p_{4}s)s}{(p_3+p_{4}s)(p_3-p_{4}s)s}=\frac{\rho }{s}+\sigma ,$$
ⓘ Symbols: $r(s,t)$: rational function, $\rho (t)$: rational function, $\sigma (t)$: rational function and $p_j$: polynomial Permalink: http://dlmf.nist.gov/19.2.E2 Encodings: TeX, pMML, png See also: Annotations for §19.2(i), §19.2 and Ch.19

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

19.2.3		$$\int \frac{\rho (t)}{s(t)}dt.$$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral and $\rho (t)$: rational function Referenced by: §19.14(ii), §19.16(ii), §19.29(ii) Permalink: http://dlmf.nist.gov/19.2.E3 Encodings: TeX, pMML, png See also: Annotations for §19.2(i), §19.2 and Ch.19

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

19.2.4		$F(\varphi ,k)$	$={\displaystyle {\int }_0^{\varphi }}{\displaystyle \frac{d\theta }{\sqrt{1-k^2{\sin }^2\theta }}}={\displaystyle {\int }_0^{\sin \varphi }}{\displaystyle \frac{dt}{\sqrt{1-t^2}\sqrt{1-k^2{t}^2}}},$
ⓘ Defines: $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind Symbols: $dx$: differential of $x$, $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument and $k$: real or complex modulus Referenced by: §19.5, §19.6(ii) Permalink: http://dlmf.nist.gov/19.2.E4 Encodings: TeX, pMML, png See also: Annotations for §19.2(ii), §19.2 and Ch.19
19.2.5		$E(\varphi ,k)$	$={\displaystyle {\int }_0^{\varphi }}\sqrt{1-k^2{\sin }^2\theta }d\theta ={\displaystyle {\int }_0^{\sin \varphi }}{\displaystyle \frac{\sqrt{1-k^2{t}^2}}{\sqrt{1-t^2}}}dt.$
ⓘ Defines: $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind Symbols: $dx$: differential of $x$, $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument and $k$: real or complex modulus Referenced by: §19.6(iii), §22.16(ii) Permalink: http://dlmf.nist.gov/19.2.E5 Encodings: TeX, pMML, png See also: Annotations for §19.2(ii), §19.2 and Ch.19
19.2.6		$D(\varphi ,k)$	$={\displaystyle {\int }_0^{\varphi }}{\displaystyle \frac{{\sin }^2\theta d\theta }{\sqrt{1-k^2{\sin }^2\theta }}}={\displaystyle {\int }_0^{\sin \varphi }}{\displaystyle \frac{t^{2}dt}{\sqrt{1-t^2}\sqrt{1-k^2{t}^2}}}=(F(\varphi ,k)-E(\varphi ,k))/k^2.$
ⓘ Defines: $D(\varphi ,k)$: incomplete elliptic integral of Legendre’s type Symbols: $dx$: differential of $x$, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument and $k$: real or complex modulus Referenced by: §19.25(i), §19.36(i) Permalink: http://dlmf.nist.gov/19.2.E6 Encodings: TeX, pMML, png See also: Annotations for §19.2(ii), §19.2 and Ch.19

19.2.7		$$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)={\int }_0^{\varphi }\frac{d\theta }{\sqrt{1-k^2{\sin }^2\theta }(1-{\alpha }^2{\sin }^2\theta )}={\int }_0^{\sin \varphi }\frac{dt}{\sqrt{1-t^2}\sqrt{1-k^2{t}^2}(1-{\alpha }^2{t}^2)}.$$
ⓘ Defines: $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind Symbols: $dx$: differential of $x$, $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter Referenced by: §19.5 Permalink: http://dlmf.nist.gov/19.2.E7 Encodings: TeX, pMML, png See also: Annotations for §19.2(ii), §19.2 and Ch.19

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:

19.2.8		$K\left(k\right)$	$=F(\pi /2,k),$
		$E\left(k\right)$	$=E(\pi /2,k),$
		$D\left(k\right)$	$=D(\pi /2,k)=(K\left(k\right)-E\left(k\right))/k^2,$
		$\mathrm{\Pi }({\alpha }^2,k)$	$=\mathrm{\Pi }(\pi /2,{\alpha }^2,k),$
ⓘ Defines: $D\left(k\right)$: complete elliptic integral of Legendre’s type, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind and $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $D(\varphi ,k)$: incomplete elliptic integral of Legendre’s type, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter Permalink: http://dlmf.nist.gov/19.2.E8 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §19.2(ii), §19.2 and Ch.19

19.2.9		$K^{\prime }\left(k\right)$	$=K\left(k^{\prime }\right),$
		$E^{\prime }\left(k\right)$	$=E\left(k^{\prime }\right),$
		$k^{\prime }$	$=\sqrt{1-k^2}.$
ⓘ Defines: $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind and $E^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the second kind Symbols: $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $k$: real or complex modulus and $k^{\prime }$: complementary modulus Referenced by: §22.11 Permalink: http://dlmf.nist.gov/19.2.E9 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.2(ii), §19.2 and Ch.19

If $m$ is an integer, then

19.2.10		$F(m\pi \pm \varphi ,k)$	$=2mK\left(k\right)\pm F(\varphi ,k),$
		$E(m\pi \pm \varphi ,k)$	$=2mE\left(k\right)\pm E(\varphi ,k),$
		$D(m\pi \pm \varphi ,k)$	$=2mD\left(k\right)\pm D(\varphi ,k).$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $D\left(k\right)$: complete elliptic integral of Legendre’s type, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $D(\varphi ,k)$: incomplete elliptic integral of Legendre’s type, $m$: nonnegative integer, $\varphi$: real or complex argument and $k$: real or complex modulus Referenced by: §19.14(i) Permalink: http://dlmf.nist.gov/19.2.E10 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.2(ii), §19.2 and Ch.19

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

19.2.11		$$\mathrm{cel}(k_c,p,a,b)={\int }_0^{\pi /2}\frac{a{\cos }^2\theta +b{\sin }^2\theta }{{\cos }^2\theta +p{\sin }^2\theta }\frac{d\theta }{\sqrt{{\cos }^2\theta +k_c^2{\sin }^2\theta }},$$
ⓘ Defines: $\mathrm{cel}(k_c,p,a,b)$: Bulirsch’s complete elliptic integral Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\sin z$: sine function, $a$: real parameter, $b$: real parameter, $p$: real parameter not equal to zero and $k_c$: real or complex variable not equal to zero Referenced by: §19.2(iii) Permalink: http://dlmf.nist.gov/19.2.E11 Encodings: TeX, pMML, png See also: Annotations for §19.2(iii), §19.2 and Ch.19

19.2.11_5		$$\mathrm{el1}(x,k_c)={\int }_0^{\arctan x}\frac{1}{\sqrt{{\cos }^2\theta +k_c^2{\sin }^2\theta }}d\theta ,$$
ⓘ Defines: $\mathrm{el1}(x,k_c)$: Bulirsch’s incomplete elliptic integral of the first kind Symbols: $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\arctan z$: arctangent function, $\sin z$: sine function, $k_c$: real or complex variable not equal to zero and $x$: real or complex variable Referenced by: §19.2(iii), Erratum (V1.1.0) for Additions Permalink: http://dlmf.nist.gov/19.2.E11_5 Encodings: TeX, pMML, png Addition (effective with 1.1.0): This equation was added and made the definition for $\mathrm{el1}(x,k_c)$ (was previously (19.2.15)). Suggested 2019-09-26 by Jan Mangaldan See also: Annotations for §19.2(iii), §19.2 and Ch.19

19.2.12		$$\mathrm{el2}(x,k_c,a,b)={\int }_0^{\arctan x}\frac{a+b{\tan }^2\theta }{\sqrt{(1+{\tan }^2\theta )(1+k_c^2{\tan }^2\theta )}}d\theta .$$
ⓘ Defines: $\mathrm{el2}(x,k_c,a,b)$: Bulirsch’s incomplete elliptic integral of the second kind Symbols: $dx$: differential of $x$, $\int$: integral, $\arctan z$: arctangent function, $\tan z$: tangent function, $a$: real parameter, $b$: real parameter, $k_c$: real or complex variable not equal to zero and $x$: real or complex variable Permalink: http://dlmf.nist.gov/19.2.E12 Encodings: TeX, pMML, png See also: Annotations for §19.2(iii), §19.2 and Ch.19

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

19.2.13		$k_c$	$=k^{\prime },$
		$p$	$=1-{\alpha }^2,$
		$x$	$=\tan \varphi ,$
ⓘ Defines: $k_c$: change of variable (locally), $p$: change of variable (locally) and $x$: change of variable (locally) Symbols: $\tan z$: tangent function, $\varphi$: real or complex argument, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter Permalink: http://dlmf.nist.gov/19.2.E13 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.2(iii), §19.2 and Ch.19

special cases include

19.2.14		$K\left(k\right)$	$=\mathrm{cel}(k_c,1,1,1),$
		$E\left(k\right)$	$=\mathrm{cel}(k_c,1,1,k_c^2),$
		$D\left(k\right)$	$=\mathrm{cel}(k_c,1,0,1),$
		$(E\left(k\right)-k^{\prime }{}^{2}K\left(k\right))/k^2$	$=\mathrm{cel}(k_c,1,1,0),$
		$\mathrm{\Pi }({\alpha }^2,k)$	$=\mathrm{cel}(k_c,p,1,1),$
ⓘ Symbols: $\mathrm{cel}(k_c,p,a,b)$: Bulirsch’s complete elliptic integral, $D\left(k\right)$: complete elliptic integral of Legendre’s type, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, ${\alpha }^2$: real or complex parameter, $p$: real parameter not equal to zero and $k_c$: real or complex variable not equal to zero Permalink: http://dlmf.nist.gov/19.2.E14 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §19.2(iii), §19.2 and Ch.19

and

19.2.15		$F(\varphi ,k)$	$=\mathrm{el1}(x,k_c)=\mathrm{el2}(x,k_c,1,1),$
		$E(\varphi ,k)$	$=\mathrm{el2}(x,k_c,1,k_c^2),$
		$D(\varphi ,k)$	$=\mathrm{el2}(x,k_c,0,1).$
ⓘ Symbols: $\mathrm{el1}(x,k_c)$: Bulirsch’s incomplete elliptic integral of the first kind, $\mathrm{el2}(x,k_c,a,b)$: Bulirsch’s incomplete elliptic integral of the second kind, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $D(\varphi ,k)$: incomplete elliptic integral of Legendre’s type, $\varphi$: real or complex argument, $k$: real or complex modulus, $k_c$: real or complex variable not equal to zero and $x$: real or complex variable Referenced by: (19.2.11_5) Permalink: http://dlmf.nist.gov/19.2.E15 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.2(iii), §19.2 and Ch.19

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

19.2.16		$$\begin{array}{ll}\mathrm{el3}(x,k_c,p)& ={\int }_0^{\arctan x}\frac{d\theta }{({\cos }^2\theta +p{\sin }^2\theta )\sqrt{{\cos }^2\theta +k_c^2{\sin }^2\theta }}\\ & =\mathrm{\Pi }(\arctan x,1-p,k),\end{array}$$
$x^2\ne -1/p$.
ⓘ Defines: $\mathrm{el3}(x,k_c,p)$: Bulirsch’s incomplete elliptic integral of the third kind Symbols: $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\int$: integral, $\arctan z$: arctangent function, $\sin z$: sine function, $k$: real or complex modulus, $p$: real parameter not equal to zero, $k_c$: real or complex variable not equal to zero and $x$: real or complex variable Permalink: http://dlmf.nist.gov/19.2.E16 Encodings: TeX, pMML, png See also: Annotations for §19.2(iii), §19.2 and Ch.19

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

19.2.17		$$R_C(x,y)=\frac{1}{2}{\int }_0^{\mathrm{\infty }}\frac{dt}{\sqrt{t+x}(t+y)},$$
ⓘ Defines: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions Symbols: $dx$: differential of $x$ and $\int$: integral Referenced by: §19.10(ii) Permalink: http://dlmf.nist.gov/19.2.E17 Encodings: TeX, pMML, png See also: Annotations for §19.2(iv), §19.2 and Ch.19

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 $x>y$:

19.2.18		$$R_C(x,y)=\frac{1}{\sqrt{y-x}}\arctan \sqrt{\frac{y-x}{x}}=\frac{1}{\sqrt{y-x}}\arccos \sqrt{x/y},$$
,
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\arccos z$: arccosine function and $\arctan z$: arctangent function Referenced by: §19.2(iv) Permalink: http://dlmf.nist.gov/19.2.E18 Encodings: TeX, pMML, png See also: Annotations for §19.2(iv), §19.2 and Ch.19

19.2.19		$$R_C(x,y)=\frac{1}{\sqrt{x-y}}\mathrm{arctanh}\sqrt{\frac{x-y}{x}}=\frac{1}{\sqrt{x-y}}\ln \frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{y}},$$
.
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{arctanh}z$: inverse hyperbolic tangent function and $\ln z$: principal branch of logarithm function Referenced by: §19.26(i), Figure 19.3.2, Figure 19.3.2, §19.36(ii) Permalink: http://dlmf.nist.gov/19.2.E19 Encodings: TeX, pMML, png See also: Annotations for §19.2(iv), §19.2 and Ch.19

The Cauchy principal value is hyperbolic:

19.2.20		$$R_C(x,y)=\sqrt{\frac{x}{x-y}}{R}_C(x-y,-y)=\frac{1}{\sqrt{x-y}}\mathrm{arctanh}\sqrt{\frac{x}{x-y}}=\frac{1}{\sqrt{x-y}}\ln \frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{-y}},$$
.
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{arctanh}z$: inverse hyperbolic tangent function and $\ln z$: principal branch of logarithm function Referenced by: §19.2(iv), §19.20(iii), §19.25(i), Figure 19.3.2, Figure 19.3.2, §19.36(i) Permalink: http://dlmf.nist.gov/19.2.E20 Encodings: TeX, pMML, png See also: Annotations for §19.2(iv), §19.2 and Ch.19

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

19.2.21		$$R_C(x,y)={\int }_0^1{(v^{2}x+(1-v^2)y)}^{-1/2}dv,$$
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $dx$: differential of $x$ and $\int$: integral Referenced by: §19.2(iv) Permalink: http://dlmf.nist.gov/19.2.E21 Encodings: TeX, pMML, png See also: Annotations for §19.2(iv), §19.2 and Ch.19

19.2.22		$$R_C(x,y)=\frac{2}{\pi }{\int }_0^{\pi /2}{R}_C(y,x{\cos }^2\theta +y{\sin }^2\theta )d\theta .$$
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral and $\sin z$: sine function Referenced by: §19.2(iv), §19.2(iv) Permalink: http://dlmf.nist.gov/19.2.E22 Encodings: TeX, pMML, png See also: Annotations for §19.2(iv), §19.2 and Ch.19