inverse hyperbolic functions

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21: 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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22: 19.7 Connection Formulas

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19.7.8 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)+\mathrm{\Pi }(\varphi ,{\omega }^2,k)=F(\varphi ,k)+\sqrt{c}{R}_C((c-1)(c-k^2),(c-{\alpha }^2)(c-{\omega }^2)),$$ ${\alpha }^2{\omega }^2=k^2$.

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Defines:

${\omega }^2$: change of variable (locally)

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter and $c$: change of variable

Referenced by:

§19.25(i), §19.6(i)

Permalink:

http://dlmf.nist.gov/19.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.7(iii), §19.7 and Ch.19

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19.7.9 $$(k^2-{\alpha }^2)\mathrm{\Pi }(\varphi ,{\alpha }^2,k)+(k^2-{\omega }^2)\mathrm{\Pi }(\varphi ,{\omega }^2,k)=k^{2}F(\varphi ,k)-{\alpha }^2{\omega }^2\sqrt{c-1}{R}_C(c(c-k^2),(c-{\alpha }^2)(c-{\omega }^2)),$$ $(1-{\alpha }^2)(1-{\omega }^2)=1-k^2$.

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Defines:

${\omega }^2$: change of variable (locally)

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter and $c$: change of variable

Permalink:

http://dlmf.nist.gov/19.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.7(iii), §19.7 and Ch.19

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19.7.10 $$(1-{\alpha }^2)\mathrm{\Pi }(\varphi ,{\alpha }^2,k)+(1-{\omega }^2)\mathrm{\Pi }(\varphi ,{\omega }^2,k)=F(\varphi ,k)+(1-{\alpha }^2-{\omega }^2)\sqrt{c-k^2}{R}_C(c(c-1),(c-{\alpha }^2)(c-{\omega }^2)),$$ $(k^2-{\alpha }^2)(k^2-{\omega }^2)=k^2(k^2-1)$.

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Defines:

${\omega }^2$: change of variable (locally)

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter and $c$: change of variable

Permalink:

http://dlmf.nist.gov/19.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.7(iii), §19.7 and Ch.19

23: 19.23 Integral Representations

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19.23.4 $$R_F(0,y,z)=\frac{2}{\pi }{\int }_0^{\pi /2}{R}_C(y,z{\cos }^2\theta )d\theta =\frac{2}{\pi }{\int }_0^{\mathrm{\infty }}{R}_C(y{\cosh }^{2}t,z)dt.$$

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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, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function and $\int$: integral

Referenced by:

§19.23

Permalink:

http://dlmf.nist.gov/19.23.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

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19.23.5 $$R_F(x,y,z)=\frac{2}{\pi }{\int }_0^{\pi /2}{R}_C(x,y{\cos }^2\theta +z{\sin }^2\theta )d\theta ,$$ $\mathrm{\Re }y>0$, $\mathrm{\Re }z>0$,

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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, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\sin z$: sine function

Referenced by:

§19.2(iv), §19.23

Permalink:

http://dlmf.nist.gov/19.23.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

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24: 19.36 Methods of Computation

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19.36.9 $$R_F(t_0^2,t_0^2+\theta c_0^2,t_0^2+\theta a_0^2)=R_F(T^2,T^2,T^2+\theta M^2)=R_C(T^2+\theta M^2,T^2).$$

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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, $M$: limit and $T$: limit

Referenced by:

§19.15

Permalink:

http://dlmf.nist.gov/19.36.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.36(ii), §19.36 and Ch.19

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19.36.11 $$R_F(1,2,4)=R_C(T^2+M^2,T^2)=\mathrm{0.68508\hspace{0.33em}58166},$$

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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, $M$: limit and $T$: limit

Permalink:

http://dlmf.nist.gov/19.36.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.36(ii), §19.36(ii), §19.36 and Ch.19

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19.36.13 $$2R_G(t_0^2,t_0^2+\theta c_0^2,t_0^2+\theta a_0^2)=\left(t_0^2+\theta \sum _{m=0}^{\mathrm{\infty }}{2}^{m-1}{c}_m^2\right)R_C(T^2+\theta M^2,T^2)+h_0+\sum _{m=1}^{\mathrm{\infty }}{2}^m(h_m-h_{m-1}).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $m$: nonnegative integer, $M$: limit, $T$: limit and $h_n$

Permalink:

http://dlmf.nist.gov/19.36.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.36(ii), §19.36(ii), §19.36 and Ch.19

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25: 19.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►The first set of main functions treated in this chapter are Legendre’s complete integrals … ►In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by $K(\alpha )$, $E(\alpha )$, $\mathrm{\Pi }(n\backslash \alpha )$, $F(\varphi \backslash \alpha )$, $E(\varphi \backslash \alpha )$, and $\mathrm{\Pi }(n;\varphi \backslash \alpha )$, where $\alpha =\arcsin k$ and $n$ is the ${\alpha }^2$ (not related to $k$) in (19.1.1) and (19.1.2). … ►The second set of main functions treated in this chapter is … ►A third set of functions, introduced by Bulirsch (1965b, a, 1969a), is …

26: 19.9 Inequalities

… ►Inequalities for both $F(\varphi ,k)$ and $E(\varphi ,k)$ involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4). …

27: 19.16 Definitions

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19.16.6 $$R_C(x,y)=R_F(x,y,y),$$

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Symbols:

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

Referenced by:

§19.19, §19.24(ii), §19.6(ii)

Permalink:

http://dlmf.nist.gov/19.16.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(i), §19.16 and Ch.19

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19.16.18 $$R_C(x,y)=R_{-\frac{1}{2}}(\frac{1}{2},1;x,y).$$

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function and $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions

Referenced by:

§19.18(i), §19.18(ii), §19.19, §19.19, §19.2(iv)

Permalink:

http://dlmf.nist.gov/19.16.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(iii), §19.16 and Ch.19

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28: 19.8 Quadratic Transformations

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19.8.14 $$2(k^2-{\alpha }^2)\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\frac{{\omega }^2-{\alpha }^2}{1+k^{\prime }}\mathrm{\Pi }({\varphi }_1,{\alpha }_1^2,k_1)+k^{2}F(\varphi ,k)-(1+k^{\prime }){\alpha }_1^2{R}_C(c_1,c_1-{\alpha }_1^2),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, ${\alpha }^2$: real or complex parameter, $k_1$: change of variable, ${\varphi }_1$: change of variable, $\omega$: change of variable, ${\alpha }_1$: change of variable and $c_1$: change of variable

Permalink:

http://dlmf.nist.gov/19.8.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(ii), §19.8(ii), §19.8 and Ch.19

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19.8.20 $$\rho \mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\frac{4}{1+k^{\prime }}\mathrm{\Pi }({\psi }_1,{\alpha }_1^2,k_1)+(\rho -1)F(\varphi ,k)-R_C(c-1,c-{\alpha }^2),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, ${\alpha }^2$: real or complex parameter, $k_1$: change of variable, ${\psi }_1$: change of variable, $\rho$: change of variable, ${\alpha }_1^2$: change of variable and $c$: change of variable

Referenced by:

§19.36(ii)

Permalink:

http://dlmf.nist.gov/19.8.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(iii), §19.8 and Ch.19

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29: 19.22 Quadratic Transformations

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19.22.20 $$(p_{\pm }^2-p_{\mp }^2)R_J(x^2,y^2,z^2,p^2)=2(p_{\pm }^2-a^2)R_J(a^2,z_{+}^2,z_{-}^2,p_{\pm }^2)-3R_F(x^2,y^2,z^2)+3R_C(z^2,p^2),$$

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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, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind and $p_{\pm }$: modified parameter

Referenced by:

§19.22(i), §19.22(iii), §19.22(iii), §19.36(ii)

Permalink:

http://dlmf.nist.gov/19.22.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §19.22(iii), §19.22 and Ch.19

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19.22.22 $$R_C(x^2,y^2)=R_C(a^2,ay).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions

Referenced by:

§19.15, §19.22(iii), §19.22(iii), §19.26(iii)

Permalink:

http://dlmf.nist.gov/19.22.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §19.22(iii), §19.22 and Ch.19

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30: 28.26 Asymptotic Approximations for Large $q$

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28.26.3 $$\varphi =2h\sinh z-\left(m+\frac{1}{2}\right)\arctan \left(\sinh z\right).$$

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Symbols:

$\sinh z$: hyperbolic sine function, $\arctan z$: arctangent function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.8 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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