functions s(?,?;r),c(?,?;r)

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1: 19.2 Definitions

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19.2.17 $$R_C(x,y)=\frac{1}{2}{\int }_0^{\mathrm{\infty }}\frac{dt}{\sqrt{t+x}(t+y)},$$

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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:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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19.2.18 $$R_C(x,y)=\frac{1}{\sqrt{y-x}}\arctan \sqrt{\frac{y-x}{x}}=\frac{1}{\sqrt{y-x}}\arccos \sqrt{x/y},$$ ,

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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:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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19.2.19 $$R_C(x,y)=\frac{1}{\sqrt{x-y}}\mathrm{arctanh}\sqrt{\frac{x-y}{x}}=\frac{1}{\sqrt{x-y}}\ln \frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{y}},$$ .

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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:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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19.2.20 $$R_C(x,y)=\sqrt{\frac{x}{x-y}}{R}_C(x-y,-y)=\frac{1}{\sqrt{x-y}}\mathrm{arctanh}\sqrt{\frac{x}{x-y}}=\frac{1}{\sqrt{x-y}}\ln \frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{-y}},$$ .

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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:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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19.2.21 $$R_C(x,y)={\int }_0^1{(v^{2}x+(1-v^2)y)}^{-1/2}dv,$$

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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:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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2: 33.14 Definitions and Basic Properties

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§33.14(iv) Solutions $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$

►The functions $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$ are defined by …

3: 19.6 Special Cases

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19.6.8 $$F(\varphi ,1)=(\sin \varphi )R_C(1,{\cos }^2\varphi )={\mathrm{gd}}^{-1}\left(\varphi \right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

§19.10(ii), §19.10(ii)

Permalink:

http://dlmf.nist.gov/19.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.6(ii), §19.6 and Ch.19

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4: 19.17 Graphics

… ►Because the $R$-function is homogeneous, there is no loss of generality in giving one variable the value $1$ or $-1$ (as in Figure 19.3.2). For $R_F$, $R_G$, and $R_J$, which are symmetric in $x,y,z$, we may further assume that $z$ is the largest of $x,y,z$ if the variables are real, then choose $z=1$, and consider only $0\le x\le 1$ and $0\le y\le 1$. …The case $y=1$ corresponds to elementary functions. ►To view $R_F(0,y,1)$ and $2R_G(0,y,1)$ for complex $y$, put $y=1-k^2$, use (19.25.1), and see Figures 19.3.7–19.3.12. … ►To view $R_F(0,y,1)$ and $2R_G(0,y,1)$ for complex $y$, put $y=1-k^2$, use (19.25.1), and see Figures 19.3.7–19.3.12. …

5: 33.16 Connection Formulas

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§33.16(iv) $s$ and $c$ in Terms of $F_{\mathrm{\ell }}$ and $G_{\mathrm{\ell }}$ when $ϵ>0$

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§33.16(v) $s$ and $c$ in Terms of $W_{\kappa ,\mu }\left(z\right)$ when

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6: 19.1 Special Notation

… ►All derivatives are denoted by differentials, not by primes. ►The first set of main functions treated in this chapter are Legendre’s complete integrals …of the first, second, and third kinds, respectively, and Legendre’s incomplete integrals … ► $R_F(x,y,z)$, $R_G(x,y,z)$, and $R_J(x,y,z,p)$ 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. ► $R_{-a}(b_1,b_2,\mathrm{\dots },b_n;z_1,z_2,\mathrm{\dots },z_n)$ is a multivariate hypergeometric function that includes all the functions in (19.1.3). …

7: 19.20 Special Cases

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19.20.5 $$2R_G(x,y,y)=y{R}_C(x,y)+\sqrt{x}.$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_G(x,y,z)$: symmetric elliptic integral of second kind

Referenced by:

§19.20(ii)

Permalink:

http://dlmf.nist.gov/19.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(ii), §19.20 and Ch.19

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19.20.13 $$2(p-x)R_J(x,y,z,p)=3R_F(x,y,z)-3\sqrt{x}{R}_C(yz,p^2),$$ $p=x\pm \sqrt{(y-x)(z-x)}$,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind

Referenced by:

§19.20(iii), §19.22(iii)

Permalink:

http://dlmf.nist.gov/19.20.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(iii), §19.20 and Ch.19

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19.20.14 $$(q+z)R_J(x,y,z,-q)=(p-z)R_J(x,y,z,p)-3R_F(x,y,z)+3{\left(\frac{xyz}{xy+pq}\right)}^{1/2}{R}_C(xy+pq,pq),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind

Referenced by:

§19.16(i), §19.16(i), §19.20(iii), §19.20(iii), §19.21(iii), §19.36(i)

Permalink:

http://dlmf.nist.gov/19.20.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(iii), §19.20 and Ch.19

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19.20.20 $$R_D(x,y,y)=\frac{3}{2(y-x)}\left(R_C(x,y)-\frac{\sqrt{x}}{y}\right),$$ $x\ne y$, $y\ne 0$,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.20(iv), §19.22(iii)

Permalink:

http://dlmf.nist.gov/19.20.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(iv), §19.20 and Ch.19

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19.20.21 $$R_D(x,x,z)=\frac{3}{z-x}\left(R_C(z,x)-\frac{1}{\sqrt{z}}\right),$$ $x\ne z$, $xz\ne 0$.

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.20(iv)

Permalink:

http://dlmf.nist.gov/19.20.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(iv), §19.20 and Ch.19

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8: 19.10 Relations to Other Functions

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19.10.2 $$(\sinh \varphi )R_C(1,{\cosh }^2\varphi )=\mathrm{gd}\left(\varphi \right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{gd}x$: Gudermannian function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $\varphi$: real or complex argument

Referenced by:

§19.10(ii), §19.6(ii)

Permalink:

http://dlmf.nist.gov/19.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.10(ii), §19.10 and Ch.19

9: 19.27 Asymptotic Approximations and Expansions

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19.27.13 $$R_J(x,y,z,p)=\frac{3}{2\sqrt{z}p}\left(\ln \left(\frac{8z}{a+g}\right)-2R_C(1,\frac{p}{z})+O\left(\left(\frac{a}{z}+\frac{a}{p}\right)\ln \frac{p}{a}\right)\right),$$ $\max (x,y)/\min (z,p)\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\ln z$: principal branch of logarithm function, $a$ and $g$

Referenced by:

§19.12, §19.20(iii)

Permalink:

http://dlmf.nist.gov/19.27.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(v), §19.27 and Ch.19

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19.27.14 $$R_J(x,y,z,p)=\frac{3}{\sqrt{yz}}{R}_C(x,p)-\frac{6}{yz}{R}_G(0,y,z)+O\left(\frac{\sqrt{x+2p}}{yz}\right),$$ $\max (x,p)/\min (y,z)\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind

Referenced by:

§19.20(iii)

Permalink:

http://dlmf.nist.gov/19.27.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(v), §19.27 and Ch.19

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19.27.16 $$R_J(x,y,z,p)=(3/\sqrt{x})R_C({(h+p)}^2,2(b+h)p)+O\left(\frac{1}{x^{3/2}}\ln \frac{x}{b+h}\right),$$ $\max (y,z,p)/x\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\ln z$: principal branch of logarithm function, $b$ and $h$

Permalink:

http://dlmf.nist.gov/19.27.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(v), §19.27 and Ch.19

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10: 33.1 Special Notation

… ►The main functions treated in this chapter are first the Coulomb radial functions $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions $f(ϵ,\mathrm{\ell };r)$, $h(ϵ,\mathrm{\ell };r)$, $s(ϵ,\mathrm{\ell };r)$, $c(ϵ,\mathrm{\ell };r)$ (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions. … ►

$f^{(0)}(ϵ,\mathrm{\ell };r)=f(ϵ,\mathrm{\ell };r)$, $f(ϵ,\mathrm{\ell };r)=s(ϵ,\mathrm{\ell };r)$, $g(ϵ,\mathrm{\ell };r)=c(ϵ,\mathrm{\ell };r)$.