inverse hyperbolic functions

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31: 19.24 Inequalities

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19.24.15 $$R_C(x,\frac{1}{2}(y+z))\le R_F(x,y,z)\le R_C(x,\sqrt{yz}),$$ $x\ge 0$,

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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

Permalink:

http://dlmf.nist.gov/19.24.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §19.24(ii), §19.24(ii), §19.24 and Ch.19

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32: 19.28 Integrals of Elliptic Integrals

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19.28.8 $${\int }_0^{\mathrm{\infty }}{R}_J(tx,y,z,tp)dt=\frac{6}{\sqrt{p}}{R}_C(p,x)R_F(0,y,z).$$

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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, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§19.28

Permalink:

http://dlmf.nist.gov/19.28.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

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33: 13.20 Uniform Asymptotic Approximations for Large $\mu$

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13.20.9 $$\zeta \sqrt{{\zeta }^2+{\alpha }^2}+{\alpha }^2\mathrm{arcsinh}\left(\frac{\zeta }{\alpha }\right)=\frac{X}{\mu }-\frac{2\kappa }{\mu }\ln \left(\frac{X+x-2\kappa }{2\sqrt{{\mu }^2-{\kappa }^2}}\right)-2\ln \left(\frac{\mu X+2{\mu }^2-\kappa x}{x\sqrt{{\mu }^2-{\kappa }^2}}\right).$$

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Symbols:

$\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $X$

Permalink:

http://dlmf.nist.gov/13.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iii), §13.20 and Ch.13

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13.20.13 $$\zeta \sqrt{{\zeta }^2-{\alpha }^2}-{\alpha }^2\mathrm{arccosh}\left(\frac{\zeta }{\alpha }\right)=\frac{X}{\mu }-\frac{2\kappa }{\mu }\ln \left(\frac{X+x-2\kappa }{2\sqrt{{\kappa }^2-{\mu }^2}}\right)-2\ln \left(\frac{\kappa x-\mu X-2{\mu }^2}{x\sqrt{{\kappa }^2-{\mu }^2}}\right),$$ $x\ge 2\kappa +2\sqrt{{\kappa }^2-{\mu }^2}$,

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $X$

Permalink:

http://dlmf.nist.gov/13.20.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

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13.20.15 $$-\zeta \sqrt{{\zeta }^2-{\alpha }^2}-{\alpha }^2\mathrm{arccosh}\left(-\frac{\zeta }{\alpha }\right)=-\frac{X}{\mu }+\frac{2\kappa }{\mu }\ln \left(\frac{2\kappa -X-x}{2\sqrt{{\kappa }^2-{\mu }^2}}\right)+2\ln \left(\frac{\mu X+2{\mu }^2-\kappa x}{x\sqrt{{\kappa }^2-{\mu }^2}}\right),$$ ,

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $X$

Permalink:

http://dlmf.nist.gov/13.20.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

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34: 15.12 Asymptotic Approximations

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15.12.6 $$\zeta =\mathrm{arccosh}z.$$

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $z$: complex variable and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/15.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

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15.12.10 $$\zeta =\mathrm{arccosh}\left(\frac{1}{4}z-1\right),$$

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $z$: complex variable and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/15.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

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35: 19.21 Connection Formulas

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19.21.12 $$(p-x)R_J(x,y,z,p)+(q-x)R_J(x,y,z,q)=3R_F(x,y,z)-3R_C(\xi ,\eta ),$$

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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, $\xi$ and $\eta$

Referenced by:

§19.15, §19.20(iii), §19.21(iii), §19.25(i), §19.26(i)

Permalink:

http://dlmf.nist.gov/19.21.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.21(iii), §19.21 and Ch.19

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36: 19.3 Graphics

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Figure 19.3.1: $K\left(k\right)$ and $E\left(k\right)$ as functions of $k^2$ for $-2\le k^2\le 1$. … Magnify

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Figure 19.3.2: $R_C(x,1)$ and the Cauchy principal value of $R_C(x,-1)$ for $0\le x\le 5$. Both functions are asymptotic to $\ln \left(4x\right)/\sqrt{4x}$ as $x\to \mathrm{\infty }$; see (19.2.19) and (19.2.20). … Magnify

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Figure 19.3.3: $F(\varphi ,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 2$, $0\le {\sin }^2\varphi \le 1$. If ${\sin }^2\varphi =1$ ($\ge k^2$), then the function reduces to $K\left(k\right)$, becoming infinite when $k^2=1$. … Magnify 3D Help

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Figure 19.3.4: $E(\varphi ,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 2$, $0\le {\sin }^2\varphi \le 1$. If ${\sin }^2\varphi =1$ ($\ge k^2$), then the function reduces to $E\left(k\right)$, with value 1 at $k^2=1$. … Magnify 3D Help

… ►In Figures 19.3.7 and 19.3.8 for complete Legendre’s elliptic integrals with complex arguments, height corresponds to the absolute value of the function and color to the phase. …

37: 19.25 Relations to Other Functions

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19.25.16 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=-\frac{1}{3}{\omega }^2{R}_J(c-1,c-k^2,c,c-{\omega }^2)+\sqrt{\frac{(c-1)(c-k^2)}{({\alpha }^2-1)(1-{\omega }^2)}}{R}_C(c({\alpha }^2-1)(1-{\omega }^2),({\alpha }^2-c)(c-{\omega }^2)),$$ ${\omega }^2=k^2/{\alpha }^2$.

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Symbols:

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

Referenced by:

§19.25(i)

Permalink:

http://dlmf.nist.gov/19.25.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(i), §19.25 and Ch.19

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19.25.30 $$\mathrm{am}(u,k)=R_C({\mathrm{cs}}^2(u,k),{\mathrm{ns}}^2(u,k)),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function and $k$: real or complex modulus

Referenced by:

§19.25(v)

Permalink:

http://dlmf.nist.gov/19.25.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(v), §19.25 and Ch.19

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38: 19.29 Reduction of General Elliptic Integrals

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19.29.8 $${\int }_y^x\frac{a_{\alpha }+b_{\alpha }t}{a_5+b_{5}t}\frac{dt}{s(t)}=\frac{2}{3}\frac{d_{\alpha \beta }{d}_{\alpha \gamma }{d}_{\alpha \delta }}{d_{\alpha 5}}{R}_J(U_{12}^2,U_{13}^2,U_{23}^2,U_{\alpha 5}^2)+2R_C(S_{\alpha 5}^2,Q_{\alpha 5}^2),$$ $S_{\alpha 5}^2\in ℂ\setminus (-\mathrm{\infty },0)$,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $(a,b)$: open interval, $\setminus$: set subtraction, $U_n$ , $S_n$ , $Q_{\alpha \beta }$ , $d_{\alpha \beta }$ , $s(t)$: product, $\alpha$: parameter, $\beta$: parameter, $\gamma$: parameter and $\delta$: parameter

Referenced by:

§19.29(i), §19.29(ii), §19.29(ii), §19.29(ii)

Permalink:

http://dlmf.nist.gov/19.29.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.29(i), §19.29 and Ch.19

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39: 18.15 Asymptotic Approximations

… ►For asymptotic expansions of $P_n\left(\cos \theta \right)$ and $P_n\left(\cosh \xi \right)$ that are uniformly valid when $0\le \theta \le \pi -\delta$ and see §14.15(iii) with $\mu =0$ and $\nu =n$. These expansions are in terms of Bessel functions and modified Bessel functions, respectively. … ►

In Terms of Elementary Functions

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In Terms of Bessel Functions

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In Terms of Airy Functions

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40: 14.3 Definitions and Hypergeometric Representations

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14.3.23

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\varphi }_{\lambda }^{(\alpha ,\beta )}\left(t\right)$: Jacobi function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\mu$: general order, $\nu$: general degree and : real variable

Referenced by:

§14.3(iv)

Permalink:

http://dlmf.nist.gov/14.3.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iv), §14.3 and Ch.14

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