§19.2 Definitions

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

§19.2(i) General Elliptic Integrals

Keywords:: general elliptic integrals
Referenced by:: §23.1, ¶ ‣ §23.6(iv)
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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)
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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 and $\sigma(t)$ : rational function
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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)
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§19.2(ii) Legendre’s Integrals

Keywords:: Legendre’s elliptic integrals
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)
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Assume $1-{\mathop{\sin\/}\nolimits^{{2}}}\phi\in\Complex\setminus(-\infty,0]$ and $1-k^{2}{\mathop{\sin\/}\nolimits^{{2}}}\phi\in\Complex\setminus(-\infty,0]$ , except that one of them may be 0, and $1-\alpha^{2}{\mathop{\sin\/}\nolimits^{{2}}}\phi\in\Complex\setminus\{ 0\}$ . Then

19.2.4 $\mathop{F\/}\nolimits\!\left(\phi,k\right)=\int _{0}^{{\phi}}\frac{d\theta}{\sqrt{1-k^{2}{\mathop{\sin\/}\nolimits^{{2}}}\theta}}=\int _{0}^{{\mathop{\sin\/}\nolimits\phi}}\frac{dt}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}},$

Defines:: $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind
Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.5, §19.6(ii)
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19.2.5 $\mathop{E\/}\nolimits\!\left(\phi,k\right)=\int _{0}^{{\phi}}\sqrt{1-k^{2}{\mathop{\sin\/}\nolimits^{{2}}}\theta}d\theta\\ =\int _{0}^{{\mathop{\sin\/}\nolimits\phi}}\frac{\sqrt{1-k^{2}t^{2}}}{\sqrt{1-t^{2}}}dt.$

Defines:: $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind
Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.6(iii), ¶ ‣ §22.16(ii)
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19.2.6 $\mathop{D\/}\nolimits\!\left(\phi,k\right)=\int _{0}^{{\phi}}\frac{{\mathop{\sin\/}\nolimits^{{2}}}\theta d\theta}{\sqrt{1-k^{2}{\mathop{\sin\/}\nolimits^{{2}}}\theta}}=\int _{0}^{{\mathop{\sin\/}\nolimits\phi}}\frac{t^{2}dt}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}=(\mathop{F\/}\nolimits\!\left(\phi,k\right)-\mathop{E\/}\nolimits\!\left(\phi,k\right))/k^{2}.$

Defines:: $\mathop{D\/}\nolimits\!\left(\phi,k\right)$ : incomplete elliptic integral of Legendre’s type
Symbols:: $dx$ : differential of $x$ , $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.25(i), ¶ ‣ §19.36(i)
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19.2.7 $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)=\int _{0}^{{\phi}}\frac{d\theta}{\sqrt{1-k^{2}{\mathop{\sin\/}\nolimits^{{2}}}\theta}(1-\alpha^{2}{\mathop{\sin\/}\nolimits^{{2}}}\theta)}=\int _{0}^{{\mathop{\sin\/}\nolimits\phi}}\frac{dt}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}(1-\alpha^{2}t^{2})}.$

Defines:: $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : Legendre’s incomplete elliptic integral of the third kind
Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter
Referenced by:: §19.5
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The paths of integration are the line segments connecting the limits of integration. The integral for $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ is well defined if $k^{2}={\mathop{\sin\/}\nolimits^{{2}}}\phi=1$ , and the Cauchy principal value (§1.4(v)) of $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ is taken if $1-\alpha^{2}{\mathop{\sin\/}\nolimits^{{2}}}\phi$ vanishes at an interior point of the integration path. Also, if $k^{2}$ and $\alpha^{2}$ are real, then $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ 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

$\mathop{K\/}\nolimits\!\left(k\right)=\mathop{F\/}\nolimits\!\left(\pi/2,k\right),$

$\mathop{E\/}\nolimits\!\left(k\right)=\mathop{E\/}\nolimits\!\left(\pi/2,k\right),$

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

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

Defines:: $\mathop{D\/}\nolimits\!\left(k\right)$ : complete elliptic integral of Legendre’s type, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind and $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : Legendre’s complete elliptic integral of the third kind
Symbols:: $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\mathop{D\/}\nolimits\!\left(\phi,k\right)$ : incomplete elliptic integral of Legendre’s type, $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : Legendre’s incomplete elliptic integral of the third kind, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter
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19.2.9

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

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

$k^{{\prime}}=\sqrt{1-k^{2}}.$

Defines:: $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind and $\mathop{{E^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the second kind
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, $k$ : real or complex modulus and $k^{{\prime}}$ : complementary modulus
Referenced by:: §22.11
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If $m$ is an integer, then

19.2.10

$\mathop{F\/}\nolimits\!\left(m\pi\pm\phi,k\right)=2m\mathop{K\/}\nolimits\!\left(k\right)\pm\mathop{F\/}\nolimits\!\left(\phi,k\right),$

$\mathop{E\/}\nolimits\!\left(m\pi\pm\phi,k\right)=2m\mathop{E\/}\nolimits\!\left(k\right)\pm\mathop{E\/}\nolimits\!\left(\phi,k\right),$

$\mathop{D\/}\nolimits\!\left(m\pi\pm\phi,k\right)=2m\mathop{D\/}\nolimits\!\left(k\right)\pm\mathop{D\/}\nolimits\!\left(\phi,k\right).$

§19.2(iii) Bulirsch’s Integrals

Notes:: See Bulirsch (1965b, 1969a, 1969b) and Bulirsch and Stoer (1968).
Keywords:: Bartky’s transformation, Bulirsch’s elliptic integrals
Referenced by:: §19.25(ii), ¶ ‣ §19.36(ii)
Permalink:: http://dlmf.nist.gov/19.2.iii

Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). Two are defined by

19.2.11 $\mathop{\mathrm{cel}\/}\nolimits\!\left(k_{c},p,a,b\right)=\int _{0}^{{\pi/2}}\frac{a{\mathop{\cos\/}\nolimits^{{2}}}\theta+b{\mathop{\sin\/}\nolimits^{{2}}}\theta}{{\mathop{\cos\/}\nolimits^{{2}}}\theta+p{\mathop{\sin\/}\nolimits^{{2}}}\theta}\frac{d\theta}{\sqrt{{\mathop{\cos\/}\nolimits^{{2}}}\theta+k_{c}^{2}{\mathop{\sin\/}\nolimits^{{2}}}\theta}},$

Defines:: $\mathop{\mathrm{cel}\/}\nolimits\!\left(k_{c},p,a,b\right)$ : Bulirsch’s complete elliptic integral
Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $k$ : real or complex modulus
Referenced by:: §19.2(iii)
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19.2.12 $\mathop{\mathrm{el2}\/}\nolimits\!\left(x,k_{c},a,b\right)=\int _{0}^{{\mathop{\mathrm{arctan}\/}\nolimits x}}\frac{a+b{\mathop{\tan\/}\nolimits^{{2}}}\theta}{\sqrt{(1+{\mathop{\tan\/}\nolimits^{{2}}}\theta)(1+k_{c}^{2}{\mathop{\tan\/}\nolimits^{{2}}}\theta)}}d\theta.$

Defines:: $\mathop{\mathrm{el2}\/}\nolimits\!\left(x,k_{c},a,b\right)$ : Bulirsch’s incomplete elliptic integral of the second kind
Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\tan\/}\nolimits z$ : tangent function and $k$ : real or complex modulus
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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 $-\infty<p<0$ , then the integral in (19.2.11) is a Cauchy principal value.

With

19.2.13

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

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

$x=\mathop{\tan\/}\nolimits\phi,$

Symbols:: $\mathop{\tan\/}\nolimits z$ : tangent function, $\phi$ : real or complex argument, $k$ : real or complex modulus, $k^{{\prime}}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter
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special cases include

19.2.14

$\mathop{K\/}\nolimits\!\left(k\right)=\mathop{\mathrm{cel}\/}\nolimits\!\left(k_{c},1,1,1\right),$

$\mathop{E\/}\nolimits\!\left(k\right)=\mathop{\mathrm{cel}\/}\nolimits\!\left(k_{c},1,1,k_{c}^{2}\right),$

$\mathop{D\/}\nolimits\!\left(k\right)=\mathop{\mathrm{cel}\/}\nolimits\!\left(k_{c},1,0,1\right),$

$(\mathop{E\/}\nolimits\!\left(k\right)-{k^{{\prime}}}^{2}\mathop{K\/}\nolimits\!\left(k\right))/k^{2}=\mathop{\mathrm{cel}\/}\nolimits\!\left(k_{c},1,1,0\right),$

$\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)=\mathop{\mathrm{cel}\/}\nolimits\!\left(k_{c},p,1,1\right),$

and

19.2.15

$\mathop{F\/}\nolimits\!\left(\phi,k\right)=\mathop{\mathrm{el1}\/}\nolimits\!\left(x,k_{c}\right)=\mathop{\mathrm{el2}\/}\nolimits\!\left(x,k_{c},1,1\right),$

$\mathop{E\/}\nolimits\!\left(\phi,k\right)=\mathop{\mathrm{el2}\/}\nolimits\!\left(x,k_{c},1,k_{c}^{2}\right),$

$\mathop{D\/}\nolimits\!\left(\phi,k\right)=\mathop{\mathrm{el2}\/}\nolimits\!\left(x,k_{c},0,1\right).$

Defines:: $\mathop{\mathrm{el1}\/}\nolimits\!\left(x,k_{c}\right)$ : Bulirsch’s incomplete elliptic integral of the first kind
Symbols:: $\mathop{\mathrm{el2}\/}\nolimits\!\left(x,k_{c},a,b\right)$ : Bulirsch’s incomplete elliptic integral of the second kind, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\mathop{D\/}\nolimits\!\left(\phi,k\right)$ : incomplete elliptic integral of Legendre’s type, $\phi$ : real or complex argument and $k$ : real or complex modulus
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The integrals are complete if $x=\infty$ . If $1<k\leq 1/\mathop{\sin\/}\nolimits\phi$ , then $k_{c}$ is pure imaginary.

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

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

Defines:: $\mathop{\mathrm{el3}\/}\nolimits\!\left(x,k_{c},p\right)$ : Bulirsch’s incomplete elliptic integral of the third kind
Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\int$ : integral, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\sin\/}\nolimits z$ : sine function and $k$ : real or complex modulus
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§19.2(iv) A Related Function: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$

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 $\mathop{R_{C}\/}\nolimits$ by Carlson (1977b, (9.8-4)) with $B=C$ . 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 $\mathop{\cos\/}\nolimits\theta=v$ in (19.23.8). For (19.2.22) put $z=x$ in (19.23.5) and interchange $x$ and $y$ .
Keywords:: $\mathop{R_{C}\/}\nolimits$ -function
Referenced by:: §19.16(i), Version 1.0.1 (June 27, 2011)
Permalink:: http://dlmf.nist.gov/19.2.iv
Clarification (effective with 1.0.1):: The title was amplified to indicate $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ is not an elliptic integral; see 19.16(iii).
Reported 2010-11-23

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

19.2.17 $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)=\frac{1}{2}\int _{0}^{{\infty}}\frac{dt}{\sqrt{t+x}(t+y)},$

Defines:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions
Symbols:: $dx$ : differential of $x$ and $\int$ : integral
Referenced by:: §19.10(ii)
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where the Cauchy principal value is taken if $y<0$ . Formulas involving $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ .

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

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

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function and $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function
Referenced by:: §19.2(iv)
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19.2.19 $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)=\frac{1}{\sqrt{x-y}}\mathop{\mathrm{arctanh}\/}\nolimits\sqrt{\frac{x-y}{x}}=\frac{1}{\sqrt{x-y}}\mathop{\ln\/}\nolimits\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{y}},$ $0<y<x$ .

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{\mathrm{arctanh}\/}\nolimits z$ : inverse hyperbolic tangent function and $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function
Referenced by:: §19.26(i), Figure 19.3.2, Figure 19.3.2, ¶ ‣ §19.36(ii)
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The Cauchy principal value is hyperbolic:

19.2.20 $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)=\sqrt{\frac{x}{x-y}}\mathop{R_{C}\/}\nolimits\!\left(x-y,-y\right)=\frac{1}{\sqrt{x-y}}\mathop{\mathrm{arctanh}\/}\nolimits\sqrt{\frac{x}{x-y}}=\frac{1}{\sqrt{x-y}}\mathop{\ln\/}\nolimits\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{-y}},$ $y<0\leq x$ .

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{\mathrm{arctanh}\/}\nolimits z$ : inverse hyperbolic tangent function and $\mathop{\ln\/}\nolimits z$ : 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)
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If the line segment with endpoints $x$ and $y$ lies in $\Complex\setminus(-\infty,0]$ , then

19.2.21 $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)=\int _{0}^{1}(v^{2}x+(1-v^{2})y)^{{-1/2}}dv,$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $dx$ : differential of $x$ and $\int$ : integral
Referenced by:: §19.2(iv)
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19.2.22 $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)=\frac{2}{\pi}\int _{0}^{{\pi/2}}\mathop{R_{C}\/}\nolimits\!\left(y,x{\mathop{\cos\/}\nolimits^{{2}}}\theta+y{\mathop{\sin\/}\nolimits^{{2}}}\theta\right)d\theta.$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral and $\mathop{\sin\/}\nolimits z$ : sine function
Referenced by:: §19.2(iv), §19.2(iv)
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§19.2 Definitions

Contents

§19.2(i) General Elliptic Integrals

§19.2(ii) Legendre’s Integrals

§19.2(iii) Bulirsch’s Integrals

§19.2(iv) A Related Function:

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