inverse trigonometric functions

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41: 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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42: 15.12 Asymptotic Approximations

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$\mathrm{\Re }z>\frac{1}{2}$ and ${\alpha }_{-}-\frac{1}{2}\pi +\delta \le \mathrm{ph}c\le {\alpha }_{+}+\frac{1}{2}\pi -\delta$, where

15.12.1 $${\alpha }_{\pm }=\arctan \left(\frac{\mathrm{ph}z-\mathrm{ph}\left(1-z\right)\mp \pi }{\ln |1-z^{-1}|}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable and ${\alpha }_{\pm }$

Permalink:

http://dlmf.nist.gov/15.12.E1

Encodings:

pMML, png, TeX

See also:

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

with $z$ restricted so that $\pm {\alpha }_{\pm }\in [0,\frac{1}{2}\pi )$.

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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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43: 25.14 Lerch’s Transcendent

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25.14.6 $$\mathrm{\Phi }(z,s,a)=\frac{1}{2}{a}^{-s}+{\int }_0^{\mathrm{\infty }}\frac{z^x}{{(a+x)}^s}dx-2{\int }_0^{\mathrm{\infty }}\frac{\sin \left(x\ln z-s\arctan \left(x/a\right)\right)}{{(a^2+x^2)}^{s/2}({\mathrm{e}}^{2\pi x}-1)}dx,$$ $\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$.

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Symbols:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Erdélyi et al. (1953a, (1.11.4), p. 28)

Notes:

For the case see Erdélyi et al. (1953a, (1.11.4), p. 28). In the case $\left|z\right|=1$ one checks that the first integral converges absolutely iff $\mathrm{\Re }s>1$.

Referenced by:

Erratum (V1.1.4) for Equation (25.14.6)

Permalink:

http://dlmf.nist.gov/25.14.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The constraint which originally read “$\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $|z|=1,\mathrm{\Re }a>0$” has been improved to be “$\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$”.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.14(ii), §25.14 and Ch.25

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44: 3.11 Approximation Techniques

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3.11.6 $$T_n\left(x\right)=\cos \left(n\arccos x\right),$$ $-1\le x\le 1$.

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $\cos z$: cosine function and $\arccos z$: arccosine function

Referenced by:

§3.11(ii)

Permalink:

http://dlmf.nist.gov/3.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §3.11(ii), §3.11 and Ch.3

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45: 30.15 Signal Analysis

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30.15.11 $$\arccos \sqrt{\mathrm{B}}+\arccos \sqrt{\alpha }=\arccos \sqrt{{\mathrm{\Lambda }}_0},$$

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Symbols:

$\arccos z$: arccosine function, $\mathrm{\Lambda }$ and $\mathrm{B}$: maximum bound

Permalink:

http://dlmf.nist.gov/30.15.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §30.15(v), §30.15 and Ch.30

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46: 5.4 Special Values and Extrema

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5.4.20 $$x_n=-n+\frac{1}{\pi }\arctan \left(\frac{\pi }{\ln n}\right)+O\left(\frac{1}{n{(\ln n)}^2}\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $x$: real variable

A&S Ref:

6.3.20 (The error estimate has been improved.)

Referenced by:

§5.4(iii)

Permalink:

http://dlmf.nist.gov/5.4.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(iii), §5.4 and Ch.5

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47: 22.20 Methods of Computation

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22.20.4 $${\varphi }_{n-1}=\frac{1}{2}\left({\varphi }_n+\arcsin \left(\frac{c_n}{a_n}\sin {\varphi }_n\right)\right),$$

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Symbols:

$\arcsin z$: arcsine function, $\sin z$: sine function, $a_n$: numbers, $c_n$: numbers, $n$: positive and ${\varphi }_N$: approximations

Referenced by:

§22.20(ii)

Permalink:

http://dlmf.nist.gov/22.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.20(ii), §22.20 and Ch.22

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48: 19.9 Inequalities

… ►Throughout this subsection we assume that , $0\le \varphi \le \pi /2$, and $\mathrm{\Delta }=\sqrt{1-k^2{\sin }^2\varphi }>0$. … ►where ${\mathrm{gd}}^{-1}\left(\varphi \right)$ is given by (4.23.41) and (4.23.42). … ►(19.9.15) is useful when $k^2$ and ${\sin }^2\varphi$ are both close to $1$, since the bounds are then nearly equal; otherwise (19.9.14) is preferable. ►Inequalities for both $F(\varphi ,k)$ and $E(\varphi ,k)$ involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4). … ►

$U=\frac{1}{2}\mathrm{arctanh}\left(\sin \varphi \right)+\frac{1}{2}{k}^{-1}\mathrm{arctanh}\left(k\sin \varphi \right).$

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49: 10.20 Uniform Asymptotic Expansions for Large Order

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10.20.3 $$\frac{2}{3}{(-\zeta )}^{\frac{3}{2}}={\int }_1^z\frac{\sqrt{t^2-1}}{t}dt=\sqrt{z^2-1}-\mathrm{arcsec}z,$$ ,

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\mathrm{arcsec}z$: arcsecant function, $z$: complex variable and $\zeta (z)$: solution

A&S Ref:

9.3.39

Referenced by:

§10.20(ii), §10.21(viii)

Permalink:

http://dlmf.nist.gov/10.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20 and Ch.10

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50: 32.11 Asymptotic Approximations for Real Variables

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32.11.27 $$\sigma =(2/\pi )\arcsin \left(\pi \lambda \right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\arcsin z$: arcsine function and $\sigma$: constant

Permalink:

http://dlmf.nist.gov/32.11.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §32.11(iv), §32.11 and Ch.32

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