19 Elliptic Integrals Legendre’s Integrals 19.1 Special Notation 19.3 Graphics

§19.2 Definitions

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Permalink:: http://dlmf.nist.gov/19.2
See also:: Annotations for Ch.19

§19.2(i) General Elliptic Integrals
§19.2(ii) Legendre’s Integrals
§19.2(iii) Bulirsch’s Integrals
§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

§19.2(i) General Elliptic Integrals

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Keywords:: general elliptic integrals
Referenced by:: §23.1, §23.6(iv)
Permalink:: http://dlmf.nist.gov/19.2.i
See also:: Annotations for §19.2 and Ch.19

Let $s^{2}(t)$ 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 $\int r(s,t)\,\mathrm{d}t$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and : 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 $r(s,t)=\frac{(p_{1}+p_{2}s)(p_{3}-p_{4}s)s}{(p_{3}+p_{4}s)(p_{3}-p_{4}s)s}=% \frac{\rho}{s}+\sigma,$

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Symbols:: : 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 while $\rho$ and $\sigma$ are rational functions of . Thus the elliptic part of (19.2.1) is

19.2.3 $\int\frac{\rho(t)}{s(t)}\,\mathrm{d}t.$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : 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

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Keywords:: Legendre’s elliptic integrals, circular cases, complete, hyperbolic cases
Referenced by:: §14.5(v), §19.4(i), §19.7(iii), §19.9(ii), §20.11(iii), §20.9(i), §22.1, §22.11, §22.16(ii), §22.16(ii), §22.16(ii), §22.16(iii), §22.16(i), §22.2, §23.6(ii), §28.6(i), §29.1, §29.2(i), §31.2(iv), §31.7(ii), Erratum (V1.1.10) for Subsection 19.2(ii) and Equation (19.2.9)
Permalink:: http://dlmf.nist.gov/19.2.ii
Clarification (effective with 1.1.10):: The material surrounding (19.2.8), (19.2.9) has been updated so that the complementary complete elliptic integrals of the first and second kind are defined with consistent multivalued properties and correct analytic continuation. Equations (19.2.8_1), (19.2.8_2) were added.
See also:: Annotations for §19.2 and Ch.19

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 $F\left(\phi,k\right)=\int_{0}^{\phi}\frac{\,\mathrm{d}\theta}{\sqrt{1-k^{2}{% \sin}^{2}\theta}}=\int_{0}^{\sin\phi}\frac{\,\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{% 1-k^{2}t^{2}}},$

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Defines:: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument and : 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 $E\left(\phi,k\right)=\int_{0}^{\phi}\sqrt{1-k^{2}{\sin}^{2}\theta}\,\mathrm{d}% \theta\\ =\int_{0}^{\sin\phi}\frac{\sqrt{1-k^{2}t^{2}}}{\sqrt{1-t^{2}}}\,\mathrm{d}t.$

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Defines:: $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument and : 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 $D\left(\phi,k\right)=\int_{0}^{\phi}\frac{{\sin}^{2}\theta\,\mathrm{d}\theta}{% \sqrt{1-k^{2}{\sin}^{2}\theta}}=\int_{0}^{\sin\phi}\frac{t^{2}\,\mathrm{d}t}{% \sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}=(F\left(\phi,k\right)-E\left(\phi,k\right))% /k^{2}.$

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Defines:: $D\left(\NVar{\phi},\NVar{k}\right)$ : incomplete elliptic integral of Legendre’s type
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $F\left(\NVar{\phi},\NVar{k}\right)$ : 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 : 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 $\Pi\left(\phi,\alpha^{2},k\right)=\int_{0}^{\phi}\frac{\,\mathrm{d}\theta}{% \sqrt{1-k^{2}{\sin}^{2}\theta}(1-\alpha^{2}{\sin}^{2}\theta)}=\int_{0}^{\sin% \phi}\frac{\,\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}(1-\alpha^{2}t^{2})}.$

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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 , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, : 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

$K\left(k\right)=F\left(\pi/2,k\right),$

$E\left(k\right)=E\left(\pi/2,k\right),$

$D\left(k\right)=D\left(\pi/2,k\right)=(K\left(k\right)-E\left(k\right))/k^{2},$

$\Pi\left(\alpha^{2},k\right)=\Pi\left(\pi/2,\alpha^{2},k\right).$

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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, : 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 ${K^{\prime}}\left(k\right)=\int_{0}^{1}\frac{\,\mathrm{d}t}{\sqrt{1-t^{2}}% \sqrt{1-(1-k^{2})t^{2}}},$

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Defines:: ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and : 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 ${E^{\prime}}\left(k\right)=\int_{0}^{1}\frac{\sqrt{1-(1-k^{2})t^{2}}}{\sqrt{1-% t^{2}}}\,\mathrm{d}t,$

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Defines:: ${E^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the second kind
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and : 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 and principal branch $|\operatorname{ph}k|\leq\pi$ . Let $k^{\prime}=\sqrt{1-k^{2}}$ . Then

19.2.9

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

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, ${E^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the second kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase, : real or complex modulus and $k^{\prime}$ : complementary modulus
Referenced by:: §19.2(ii), §22.11, Erratum (V1.1.10) for Subsection 19.2(ii) and Equation (19.2.9), Erratum (V1.1.10) for Subsection 19.2(ii) and Equation (19.2.9)
Permalink:: http://dlmf.nist.gov/19.2.E9
Encodings:: TeX, TeX, pMML, pMML, png, png
Correction (effective with 1.1.10):: This equation has been updated so that it has correct analytic continuation.
See also:: Annotations for §19.2(ii), §19.2 and Ch.19

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

If is an integer, then

19.2.10

$F\left(m\pi\pm\phi,k\right)=2mK\left(k\right)\pm F\left(\phi,k\right),$

$E\left(m\pi\pm\phi,k\right)=2mE\left(k\right)\pm E\left(\phi,k\right),$

$D\left(m\pi\pm\phi,k\right)=2mD\left(k\right)\pm D\left(\phi,k\right).$

§19.2(iii) Bulirsch’s Integrals

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Keywords:: Bartky’s transformation, Bulirsch’s elliptic integrals
Notes:: See Bulirsch (1965b, 1969a, 1969b) and Bulirsch and Stoer (1968).
Referenced by:: §19.25(ii), §19.36(ii)
Permalink:: http://dlmf.nist.gov/19.2.iii
Addition (effective with 1.1.0):: Equation (19.2.11_5) was added.
Suggested 2019-09-26 by Jan Mangaldan
See also:: Annotations for §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 $\operatorname{cel}\left(k_{c},p,a,b\right)=\int_{0}^{\pi/2}\frac{a{\cos}^{2}% \theta+b{\sin}^{2}\theta}{{\cos}^{2}\theta+p{\sin}^{2}\theta}\frac{\,\mathrm{d% }\theta}{\sqrt{{\cos}^{2}\theta+k_{c}^{2}{\sin}^{2}\theta}},$

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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 , $\int$ : integral, $\sin\NVar{z}$ : sine function, : real parameter, : real parameter, : 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 $\operatorname{el1}\left(x,k_{c}\right)=\int_{0}^{\operatorname{arctan}x}\frac{% 1}{\sqrt{{\cos}^{2}\theta+k_{c}^{2}{\sin}^{2}\theta}}\,\mathrm{d}\theta,$

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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 , $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\sin\NVar{z}$ : sine function, $k_{c}$ : real or complex variable not equal to zero and : 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 $\operatorname{el2}\left(x,k_{c},a,b\right)=\int_{0}^{\operatorname{arctan}x}% \frac{a+b{\tan}^{2}\theta}{\sqrt{(1+{\tan}^{2}\theta)(1+k_{c}^{2}{\tan}^{2}% \theta)}}\,\mathrm{d}\theta.$

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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 , $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\tan\NVar{z}$ : tangent function, : real parameter, : real parameter, $k_{c}$ : real or complex variable not equal to zero and : 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 are real parameters, and $k_{c}$ and are real or complex variables, with $p\neq 0$ , $k_{c}\neq 0$ . If $-\infty<p<0$ , then the integral in (19.2.11) is a Cauchy principal value.

With

19.2.13

$k_{c}=k^{\prime},$

$p=1-\alpha^{2},$

$x=\tan\phi,$

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Defines:: $k_{c}$ : change of variable (locally), : change of variable (locally) and : 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

$K\left(k\right)=\operatorname{cel}\left(k_{c},1,1,1\right),$

$E\left(k\right)=\operatorname{cel}\left(k_{c},1,1,k_{c}^{2}\right),$

$D\left(k\right)=\operatorname{cel}\left(k_{c},1,0,1\right),$

$(E\left(k\right)-{k^{\prime}}^{2}K\left(k\right))/k^{2}=\operatorname{cel}% \left(k_{c},1,1,0\right),$

$\Pi\left(\alpha^{2},k\right)=\operatorname{cel}\left(k_{c},p,1,1\right),$

and

19.2.15

$F\left(\phi,k\right)=\operatorname{el1}\left(x,k_{c}\right)=\operatorname{el2}% \left(x,k_{c},1,1\right),$

$E\left(\phi,k\right)=\operatorname{el2}\left(x,k_{c},1,k_{c}^{2}\right),$

$D\left(\phi,k\right)=\operatorname{el2}\left(x,k_{c},0,1\right).$

The integrals are complete if $x=\infty$ . If $1<k\leq 1/\sin\phi$ , then $k_{c}$ is pure imaginary.

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

19.2.16 $\operatorname{el3}\left(x,k_{c},p\right)=\int_{0}^{\operatorname{arctan}x}% \frac{\,\mathrm{d}\theta}{({\cos}^{2}\theta+p{\sin}^{2}\theta)\sqrt{{\cos}^{2}% \theta+k_{c}^{2}{\sin}^{2}\theta}}=\Pi\left(\operatorname{arctan}x,1-p,k\right),$ $x^{2}\neq-1/p$ .

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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 , $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\sin\NVar{z}$ : sine function, : real or complex modulus, : real parameter not equal to zero, $k_{c}$ : real or complex variable not equal to zero and : 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: $R_{C}\left(x,y\right)$

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Keywords:: $R_{C}$ -function
Notes:: To prove (19.2.20) evaluate the two parts of the Cauchy principal value (intervals $(0,-y-\delta)$ and $(-y+\delta,\infty)$ ) using Carlson (1977b, (8.2-2)), and reduce the first part to $R_{C}$ by Carlson (1977b, (9.8-4)) with . Apply (19.12.7) to both parts as $\delta\to 0$ and combine the two logarithms. For (19.2.21) see (19.16.18) and put $\cos\theta=v$ in (19.23.8). For (19.2.22) put in (19.23.5) and interchange and .
Referenced by:: §19.16(i), Erratum (V1.0.1) for Clarifications, Erratum (V1.0.7) for Usability
Permalink:: http://dlmf.nist.gov/19.2.iv
Addition (effective with 1.0.7):: The cross-reference to (19.6.15) has been added after (19.2.20).
Suggested 2013-08-26 by Roahn Wynar
Clarification (effective with 1.0.1):: The title was amplified to indicate $R_{C}\left(x,y\right)$ is not an elliptic integral; see 19.16(iii).
See also:: Annotations for §19.2 and Ch.19

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

19.2.17 $R_{C}\left(x,y\right)=\frac{1}{2}\int_{0}^{\infty}\frac{\,\mathrm{d}t}{\sqrt{t% +x}(t+y)},$

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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 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 $\Pi\left(\phi,\alpha^{2},k\right)$ 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 and are positive, $R_{C}\left(x,y\right)$ is an inverse circular function if and an inverse hyperbolic function (or logarithm) if :

19.2.18 $R_{C}\left(x,y\right)=\frac{1}{\sqrt{y-x}}\operatorname{arctan}\sqrt{\frac{y-x% }{x}}=\frac{1}{\sqrt{y-x}}\operatorname{arccos}\sqrt{x/y},$ $0\leq x<y$ ,

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

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

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