inverse trigonometric functions

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31: 36.13 Kelvin’s Ship-Wave Pattern

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36.13.5 $$|\varphi |={\varphi }_c=\arcsin \left(\frac{1}{3}\right)={19}^{\circ }.47122.$$

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Symbols:

$\arcsin z$: arcsine function

Permalink:

http://dlmf.nist.gov/36.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.13 and Ch.36

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32: 18.15 Asymptotic Approximations

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18.15.18 $$\xi =\frac{1}{2}\left(\sqrt{x-x^2}+\arcsin (\sqrt{x})\right),$$ $0\le x\le 1$.

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Defines:

$\xi$ (locally)

Symbols:

$\arcsin z$: arcsine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.15.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(iv), §18.15(iv), §18.15 and Ch.18

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33: 14.20 Conical (or Mehler) Functions

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14.20.20 $$\sigma (\mu ,\tau )=\frac{\exp \left(\mu -\tau \arctan \alpha \right)}{{\left({\mu }^2+{\tau }^2\right)}^{\mu /2}}.$$

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Symbols:

$\exp z$: exponential function, $\arctan z$: arctangent function, $\tau$: real variable, $\mu$: general order, $\alpha$ and $\sigma (\mu ,\tau )$

Permalink:

http://dlmf.nist.gov/14.20.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(viii), §14.20 and Ch.14

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14.20.21 $${\left({\alpha }^2+\eta \right)}^{1/2}+\frac{1}{2}\alpha \ln \eta -\alpha \ln \left({\left({\alpha }^2+\eta \right)}^{1/2}+\alpha \right)=\arccos \left(\frac{x}{{\left(1+{\alpha }^2\right)}^{1/2}}\right)+\frac{\alpha }{2}\ln \left(\frac{1+{\alpha }^2+\left({\alpha }^2-1\right)x^2-2\alpha x{\left(1+{\alpha }^2-x^2\right)}^{1/2}}{\left(1+{\alpha }^2\right)\left(1-x^2\right)}\right),$$

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Symbols:

$\arccos z$: arccosine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $\eta$

Permalink:

http://dlmf.nist.gov/14.20.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(viii), §14.20 and Ch.14

►where the inverse trigonometric functions take their principal values. … ►

14.20.22 $${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\frac{\beta \exp \left(\mu \beta \arctan \beta \right)}{\mathrm{\Gamma }\left(\mu +1\right){\left(1+{\beta }^2\right)}^{\mu /2}}\frac{{\mathrm{e}}^{-\mu \rho }}{{\left(1+{\beta }^2-x^2{\beta }^2\right)}^{1/4}}\left(1+O\left(\frac{1}{\mu }\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\arctan z$: arctangent function, $x$: real variable, $\tau$: real variable, $\mu$: general order, $\beta$ and $\rho$

Referenced by:

§14.20(ix)

Permalink:

http://dlmf.nist.gov/14.20.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(ix), §14.20 and Ch.14

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14.20.24 $$\rho =\frac{1}{2}\ln \left(\frac{\left(1-{\beta }^2\right)x^2+1+{\beta }^2+2x{\left(1+{\beta }^2-{\beta }^2{x}^2\right)}^{1/2}}{1-x^2}\right)+\beta \arctan \left(\frac{\beta x}{\sqrt{1+{\beta }^2-{\beta }^2{x}^2}}\right)-\frac{1}{2}\ln \left(1+{\beta }^2\right),$$

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Symbols:

$\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\beta$ and $\rho$

Permalink:

http://dlmf.nist.gov/14.20.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(ix), §14.20 and Ch.14

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34: 6.14 Integrals

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6.14.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{si}\left(t\right)dt=-\frac{1}{a}\arctan a,$$ $\mathrm{\Re }a>0$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\mathrm{\Re }$: real part and $\mathrm{si}\left(z\right)$: sine integral

A&S Ref:

5.2.29

Permalink:

http://dlmf.nist.gov/6.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(i), §6.14 and Ch.6

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35: 33.7 Integral Representations

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33.7.3 $$H_{\mathrm{\ell }}^{-}(\eta ,\rho )=\frac{-\mathrm{i}{\mathrm{e}}^{-\pi \eta }{\rho }^{\mathrm{\ell }+1}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{\int }_0^{\mathrm{\infty }}\left(\frac{\exp \left(-\mathrm{i}(\rho \tanh t-2\eta t)\right)}{{(\cosh t)}^{2\mathrm{\ell }+2}}+\mathrm{i}{(1+t^2)}^{\mathrm{\ell }}\exp \left(-\rho t+2\eta \arctan t\right)\right)dt,$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\arctan z$: arctangent function, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

A&S Ref:

14.3.3

Referenced by:

§33.7

Permalink:

http://dlmf.nist.gov/33.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.7 and Ch.33

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36: 19.25 Relations to Other Functions

§19.25 Relations to Other Functions

… ►then the five nontrivial permutations of $x,y,z$ that leave $R_F$ invariant change $k^2$ ($=(z-y)/(z-x)$) into $1/k^2$, $k_{}^{\prime }{}_{}{}^2$, $1/k_{}^{\prime }{}_{}{}^2$, $-k^2/k_{}^{\prime }{}_{}{}^2$, $-k_{}^{\prime }{}_{}{}^2/k^2$, and $\sin \varphi$ ($=\sqrt{(z-x)/z}$) into $k\sin \varphi$, $-\mathrm{i}\tan \varphi$, $-\mathrm{i}{k}^{\prime }\tan \varphi$, $(k^{\prime }\sin \varphi )/\sqrt{1-k^2{\sin }^2\varphi }$, $-\mathrm{i}k\sin \varphi /\sqrt{1-k^2{\sin }^2\varphi }$. … ►Then … ►Inversions of 12 elliptic integrals of the first kind, producing the 12 Jacobian elliptic functions, are combined and simplified by using the properties of $R_F(x,y,z)$. … …

37: 22.15 Inverse Functions

… ►The inverse Jacobian elliptic functions can be defined in an analogous manner to the inverse trigonometric functions (§4.23). …

38: 1.14 Integral Transforms

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Inversion

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Inversion

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Inversion

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Inversion

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Inversion

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39: 25.5 Integral Representations

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25.5.10 $$\zeta \left(s\right)=\frac{2^{s-1}}{1-2^{1-s}}{\int }_0^{\mathrm{\infty }}\frac{\cos \left(s\arctan x\right)}{{(1+x^2)}^{s/2}\cosh \left(\frac{1}{2}\pi x\right)}dx.$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\arctan z$: arctangent function, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Sources:

Lindelöf (1905, (6), p. 98); with $x=2t$; Erdélyi et al. (1953a, (1.12.15), p. 33)

Permalink:

http://dlmf.nist.gov/25.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(i), §25.5 and Ch.25

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25.5.11 $$\zeta \left(s\right)=\frac{1}{2}+\frac{1}{s-1}+2{\int }_0^{\mathrm{\infty }}\frac{\sin \left(s\arctan x\right)}{{(1+x^2)}^{s/2}({\mathrm{e}}^{2\pi x}-1)}dx.$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\pi$: 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, $\sin z$: sine function, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Sources:

Lindelöf (1905, (3), p. 97); Erdélyi et al. (1953a, (1.12.12), p. 33)

Permalink:

http://dlmf.nist.gov/25.5.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(i), §25.5 and Ch.25

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25.5.12 $$\zeta \left(s\right)=\frac{2^{s-1}}{s-1}-2^s{\int }_0^{\mathrm{\infty }}\frac{\sin \left(s\arctan x\right)}{{(1+x^2)}^{s/2}({\mathrm{e}}^{\pi x}+1)}dx.$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\pi$: 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, $\sin z$: sine function, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Source:

Lindelöf (1905, (4), p. 98); with $x=2t$

Permalink:

http://dlmf.nist.gov/25.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(i), §25.5 and Ch.25

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40: 11.7 Integrals and Sums

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11.7.15 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{𝐋}_0\left(t\right)dt=\frac{2}{\pi \sqrt{a^2-1}}\arcsin \left(\frac{1}{a}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arcsin z$: arcsine function, ${𝐋}_{\nu }\left(z\right)$: modified Struve function and $a$: parameter

Permalink:

http://dlmf.nist.gov/11.7.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(iii), §11.7 and Ch.11

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