# §19.2 Definitions

## §19.2(i) General Elliptic Integrals

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 $\int r(s,t)\,\mathrm{d}t$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: 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 $\int\frac{\rho(t)}{s(t)}\,\mathrm{d}t.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: 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

## §19.2(ii) Legendre’s Integrals

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

 19.2.4 $\displaystyle F\left(\phi,k\right)$ $\displaystyle=\int_{0}^{\phi}\frac{\,\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin}^{2}% \theta}}=\int_{0}^{\sin\phi}\frac{\,\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^% {2}}},$ ⓘ Defines: $F\left(\NVar{\phi},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the first kind Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\sin\NVar{z}$: sine function, $\phi$: 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 $\displaystyle E\left(\phi,k\right)$ $\displaystyle=\int_{0}^{\phi}\sqrt{1-k^{2}{\sin}^{2}\theta}\,\mathrm{d}\theta=% \int_{0}^{\sin\phi}\frac{\sqrt{1-k^{2}t^{2}}}{\sqrt{1-t^{2}}}\,\mathrm{d}t.$ ⓘ Defines: $E\left(\NVar{\phi},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the second kind Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\sin\NVar{z}$: sine function, $\phi$: 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 $\displaystyle D\left(\phi,k\right)$ $\displaystyle=\int_{0}^{\phi}\frac{{\sin}^{2}\theta\,\mathrm{d}\theta}{\sqrt{1% -k^{2}{\sin}^{2}\theta}}=\int_{0}^{\sin\phi}\frac{t^{2}\,\mathrm{d}t}{\sqrt{1-% t^{2}}\sqrt{1-k^{2}t^{2}}}=(F\left(\phi,k\right)-E\left(\phi,k\right))/k^{2}.$ ⓘ Defines: $D\left(\NVar{\phi},\NVar{k}\right)$: incomplete elliptic integral of Legendre’s type Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $F\left(\NVar{\phi},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the first kind, $E\left(\NVar{\phi},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the second kind, $\int$: integral, $\sin\NVar{z}$: sine function, $\phi$: 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 $\Pi\left(\phi,\alpha^{2},k\right)=\int_{0}^{\phi}\frac{\,\mathrm{d}\theta}{% \sqrt{1-k^{2}{\sin}^{2}\theta}(1-\alpha^{2}{\sin}^{2}\theta)}=\int_{0}^{\sin% \phi}\frac{\,\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}(1-\alpha^{2}t^{2})}.$ ⓘ Defines: $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the third kind Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\sin\NVar{z}$: sine function, $\phi$: 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 $E\left(\phi,k\right)$ is well defined if $k^{2}={\sin}^{2}\phi=1$, and the Cauchy principal value (§1.4(v)) of $\Pi\left(\phi,\alpha^{2},k\right)$ is taken if $1-\alpha^{2}{\sin}^{2}\phi$ vanishes at an interior point of the integration path. Also, if $k^{2}$ and $\alpha^{2}$ are real, then $\Pi\left(\phi,\alpha^{2},k\right)$ 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 $\phi=\pi/2$ are the complete integrals:

 19.2.8 $\displaystyle K\left(k\right)$ $\displaystyle=F\left(\pi/2,k\right),$ $\displaystyle E\left(k\right)$ $\displaystyle=E\left(\pi/2,k\right),$ $\displaystyle D\left(k\right)$ $\displaystyle=D\left(\pi/2,k\right)=(K\left(k\right)-E\left(k\right))/k^{2},$ $\displaystyle\Pi\left(\alpha^{2},k\right)$ $\displaystyle=\Pi\left(\pi/2,\alpha^{2},k\right).$ ⓘ Defines: $D\left(\NVar{k}\right)$: complete elliptic integral of Legendre’s type, $K\left(\NVar{k}\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$: Legendre’s complete elliptic integral of the second kind and $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$: Legendre’s complete elliptic integral of the third kind Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $F\left(\NVar{\phi},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the first kind, $E\left(\NVar{\phi},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the second kind, $D\left(\NVar{\phi},\NVar{k}\right)$: incomplete elliptic integral of Legendre’s type, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the third kind, $k$: real or complex modulus and $\alpha^{2}$: real or complex parameter Referenced by: §19.2(ii), Erratum (V1.1.10) for Subsection 19.2(ii) and Equation (19.2.9) 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

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

 19.2.8_1 $\displaystyle{K^{\prime}}\left(k\right)$ $\displaystyle=\int_{0}^{1}\frac{\,\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-(1-k^{2})% t^{2}}},$ ⓘ Defines: ${K^{\prime}}\left(\NVar{k}\right)$: Legendre’s complementary complete elliptic integral of the first kind Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $k$: real or complex modulus Referenced by: §19.2(ii) Permalink: http://dlmf.nist.gov/19.2.E8_1 Encodings: TeX, pMML, png Addition (effective with 1.1.10): This equation was added. See also: Annotations for §19.2(ii), §19.2 and Ch.19 19.2.8_2 $\displaystyle{E^{\prime}}\left(k\right)$ $\displaystyle=\int_{0}^{1}\frac{\sqrt{1-(1-k^{2})t^{2}}}{\sqrt{1-t^{2}}}\,% \mathrm{d}t,$ ⓘ Defines: ${E^{\prime}}\left(\NVar{k}\right)$: Legendre’s complementary complete elliptic integral of the second kind Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $k$: real or complex modulus Referenced by: §19.2(ii) Permalink: http://dlmf.nist.gov/19.2.E8_2 Encodings: TeX, pMML, png Addition (effective with 1.1.10): This equation was added. See also: Annotations for §19.2(ii), §19.2 and Ch.19

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

 19.2.9 $\displaystyle{K^{\prime}}\left(k\right)$ $\displaystyle=\begin{cases}K\left(k^{\prime}\right),&|\operatorname{ph}k|\leq% \tfrac{1}{2}\pi,\\ K\left(k^{\prime}\right)\mp 2\mathrm{i}K\left(-k\right),&\tfrac{1}{2}\pi<\pm% \operatorname{ph}k<\pi,\end{cases}$ $\displaystyle{E^{\prime}}\left(k\right)$ $\displaystyle=\begin{cases}E\left(k^{\prime}\right),&|\operatorname{ph}k|\leq% \tfrac{1}{2}\pi,\\ E\left(k^{\prime}\right)\mp 2\mathrm{i}(K\left(-k\right)-E\left(-k\right)),&% \tfrac{1}{2}\pi<\pm\operatorname{ph}k<\pi.\end{cases}$

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

 19.2.10 $\displaystyle F\left(m\pi\pm\phi,k\right)$ $\displaystyle=2mK\left(k\right)\pm F\left(\phi,k\right),$ $\displaystyle E\left(m\pi\pm\phi,k\right)$ $\displaystyle=2mE\left(k\right)\pm E\left(\phi,k\right),$ $\displaystyle D\left(m\pi\pm\phi,k\right)$ $\displaystyle=2mD\left(k\right)\pm D\left(\phi,k\right).$

## §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)). Three are defined by

 19.2.11 $\operatorname{cel}\left(k_{c},p,a,b\right)=\int_{0}^{\pi/2}\frac{a{\cos}^{2}% \theta+b{\sin}^{2}\theta}{{\cos}^{2}\theta+p{\sin}^{2}\theta}\frac{\,\mathrm{d% }\theta}{\sqrt{{\cos}^{2}\theta+k_{c}^{2}{\sin}^{2}\theta}},$ ⓘ Defines: $\operatorname{cel}\left(\NVar{k_{c}},\NVar{p},\NVar{a},\NVar{b}\right)$: Bulirsch’s complete elliptic integral Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\sin\NVar{z}$: 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 $\operatorname{el1}\left(x,k_{c}\right)=\int_{0}^{\operatorname{arctan}x}\frac{% 1}{\sqrt{{\cos}^{2}\theta+k_{c}^{2}{\sin}^{2}\theta}}\,\mathrm{d}\theta,$ ⓘ Defines: $\operatorname{el1}\left(\NVar{x},\NVar{k_{c}}\right)$: Bulirsch’s incomplete elliptic integral of the first kind Symbols: $\cos\NVar{z}$: cosine function, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\operatorname{arctan}\NVar{z}$: arctangent function, $\sin\NVar{z}$: 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 $\operatorname{el1}\left(x,k_{c}\right)$ (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 $\operatorname{el2}\left(x,k_{c},a,b\right)=\int_{0}^{\operatorname{arctan}x}% \frac{a+b{\tan}^{2}\theta}{\sqrt{(1+{\tan}^{2}\theta)(1+k_{c}^{2}{\tan}^{2}% \theta)}}\,\mathrm{d}\theta.$ ⓘ Defines: $\operatorname{el2}\left(\NVar{x},\NVar{k_{c}},\NVar{a},\NVar{b}\right)$: Bulirsch’s incomplete elliptic integral of the second kind Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\operatorname{arctan}\NVar{z}$: arctangent function, $\tan\NVar{z}$: 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 $p\neq 0$, $k_{c}\neq 0$. If $-\infty, then the integral in (19.2.11) is a Cauchy principal value.

With

 19.2.13 $\displaystyle k_{c}$ $\displaystyle=k^{\prime},$ $\displaystyle p$ $\displaystyle=1-\alpha^{2},$ $\displaystyle x$ $\displaystyle=\tan\phi,$ ⓘ Defines: $k_{c}$: change of variable (locally), $p$: change of variable (locally) and $x$: change of variable (locally) Symbols: $\tan\NVar{z}$: tangent function, $\phi$: 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 $\displaystyle K\left(k\right)$ $\displaystyle=\operatorname{cel}\left(k_{c},1,1,1\right),$ $\displaystyle E\left(k\right)$ $\displaystyle=\operatorname{cel}\left(k_{c},1,1,k_{c}^{2}\right),$ $\displaystyle D\left(k\right)$ $\displaystyle=\operatorname{cel}\left(k_{c},1,0,1\right),$ $\displaystyle(E\left(k\right)-{k^{\prime}}^{2}K\left(k\right))/k^{2}$ $\displaystyle=\operatorname{cel}\left(k_{c},1,1,0\right),$ $\displaystyle\Pi\left(\alpha^{2},k\right)$ $\displaystyle=\operatorname{cel}\left(k_{c},p,1,1\right),$

and

 19.2.15 $\displaystyle F\left(\phi,k\right)$ $\displaystyle=\operatorname{el1}\left(x,k_{c}\right)=\operatorname{el2}\left(x% ,k_{c},1,1\right),$ $\displaystyle E\left(\phi,k\right)$ $\displaystyle=\operatorname{el2}\left(x,k_{c},1,k_{c}^{2}\right),$ $\displaystyle D\left(\phi,k\right)$ $\displaystyle=\operatorname{el2}\left(x,k_{c},0,1\right).$

The integrals are complete if $x=\infty$. If $1, then $k_{c}$ is pure imaginary.

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

 19.2.16 $\operatorname{el3}\left(x,k_{c},p\right)=\int_{0}^{\operatorname{arctan}x}% \frac{\,\mathrm{d}\theta}{({\cos}^{2}\theta+p{\sin}^{2}\theta)\sqrt{{\cos}^{2}% \theta+k_{c}^{2}{\sin}^{2}\theta}}=\Pi\left(\operatorname{arctan}x,1-p,k\right),$ $x^{2}\neq-1/p$. ⓘ Defines: $\operatorname{el3}\left(\NVar{x},\NVar{k_{c}},\NVar{p}\right)$: Bulirsch’s incomplete elliptic integral of the third kind Symbols: $\cos\NVar{z}$: cosine function, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the third kind, $\int$: integral, $\operatorname{arctan}\NVar{z}$: arctangent function, $\sin\NVar{z}$: 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

## §19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

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

 19.2.17 $R_{C}\left(x,y\right)=\frac{1}{2}\int_{0}^{\infty}\frac{\,\mathrm{d}t}{\sqrt{t% +x}(t+y)},$ ⓘ Defines: $R_{C}\left(\NVar{x},\NVar{y}\right)$: Carlson’s combination of inverse circular and inverse hyperbolic functions Symbols: $\,\mathrm{d}\NVar{x}$: 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 $y<0$. Formulas involving $\Pi\left(\phi,\alpha^{2},k\right)$ that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using $R_{C}\left(x,y\right)$.

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, $R_{C}\left(x,y\right)$ is an inverse circular function if $x and an inverse hyperbolic function (or logarithm) if $x>y$:

 19.2.18 $R_{C}\left(x,y\right)=\frac{1}{\sqrt{y-x}}\operatorname{arctan}\sqrt{\frac{y-x% }{x}}=\frac{1}{\sqrt{y-x}}\operatorname{arccos}\sqrt{x/y},$ $0\leq x, ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\operatorname{arccos}\NVar{z}$: arccosine function and $\operatorname{arctan}\NVar{z}$: 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 $R_{C}\left(x,y\right)=\frac{1}{\sqrt{x-y}}\operatorname{arctanh}\sqrt{\frac{x-% y}{x}}=\frac{1}{\sqrt{x-y}}\ln\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{y}},$ $0.

The Cauchy principal value is hyperbolic:

 19.2.20 $R_{C}\left(x,y\right)=\sqrt{\frac{x}{x-y}}R_{C}\left(x-y,-y\right)=\frac{1}{% \sqrt{x-y}}\operatorname{arctanh}\sqrt{\frac{x}{x-y}}=\frac{1}{\sqrt{x-y}}\ln% \frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{-y}},$ $y<0\leq x$.

For the special cases of $R_{C}\left(x,x\right)$ and $R_{C}\left(0,y\right)$ see (19.6.15).

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

 19.2.21 $R_{C}\left(x,y\right)=\int_{0}^{1}(v^{2}x+(1-v^{2})y)^{-1/2}\,\mathrm{d}v,$ ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\,\mathrm{d}\NVar{x}$: 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 $R_{C}\left(x,y\right)=\frac{2}{\pi}\int_{0}^{\pi/2}R_{C}\left(y,x{\cos}^{2}% \theta+y{\sin}^{2}\theta\right)\,\mathrm{d}\theta.$