trigonometric arguments

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31: Bibliography M

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Über die asymptotische Entwicklung von ${\displaystyle {\int }_0^{-\mathrm{\infty }-i(\arg w-\mu )}}{e}^{\nu z-w\sinh z}𝑑z$, für große Werte von $|w|$ und $|\nu |$. I, II. Proc. Akad. Wet. Amsterdam 35, pp. 1170–1180, 1291–1303 (German).

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External Links:

ISSN 0370-0348, ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

BibTeX

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32: 6.1 Special Notation

… ►Unless otherwise noted, primes indicate derivatives with respect to the argument. ►The main functions treated in this chapter are the exponential integrals $\mathrm{Ei}\left(x\right)$, $E_1\left(z\right)$, and $\mathrm{Ein}\left(z\right)$; the logarithmic integral $\mathrm{li}\left(x\right)$; the sine integrals $\mathrm{Si}\left(z\right)$ and $\mathrm{si}\left(z\right)$; the cosine integrals $\mathrm{Ci}\left(z\right)$ and $\mathrm{Cin}\left(z\right)$. …

33: 22.19 Physical Applications

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22.19.2 $$\sin \left(\frac{1}{2}\theta (t)\right)=\sin \left(\frac{1}{2}\alpha \right)\mathrm{sn}(t+K,\sin \left(\frac{1}{2}\alpha \right)),$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $\theta (t)$: angular displacement and $\alpha$: initial displacement

Referenced by:

§22.19(i), §22.19(i), Erratum (V1.0.8) for Equation (22.19.2)

Permalink:

http://dlmf.nist.gov/22.19.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.8):

Originally the first argument to the function $\mathrm{sn}$ was given incorrectly as $t$. The correct argument is $t+K$.

Reported 2014-03-05 by Svante Janson

See also:

Annotations for §22.19(i), §22.19 and Ch.22

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34: 10.40 Asymptotic Expansions for Large Argument

§10.40 Asymptotic Expansions for Large Argument

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Products

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§10.40(ii) Error Bounds for Real Argument and Order

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§10.40(iii) Error Bounds for Complex Argument and Order

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35: 4.37 Inverse Hyperbolic Functions

… ►The principal values (or principal branches) of the inverse $\sinh$, $\cosh$, and $\tanh$ are obtained by introducing cuts in the $z$-plane as indicated in Figure 4.37.1(i)-(iii), and requiring the integration paths in (4.37.1)–(4.37.3) not to cross these cuts. …The principal branches are denoted by $\mathrm{arcsinh}$, $\mathrm{arccosh}$, $\mathrm{arctanh}$ respectively. … ►Graphs of the principal values for real arguments are given in §4.29. This section also indicates conformal mappings, and surface plots for complex arguments. … ►For the corresponding results for $\mathrm{arccsch}z$, $\mathrm{arcsech}z$, and $\mathrm{arccoth}z$, use (4.37.7)–(4.37.9); compare §4.23(iv). …

36: 14.19 Toroidal (or Ring) Functions

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14.19.2 $$P_{\nu -\frac{1}{2}}^{\mu }\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(\frac{1}{2}-\mu \right)}{{\pi }^{1/2}{\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}𝐅(\frac{1}{2}-\mu ,\frac{1}{2}+\nu -\mu ;1-2\mu ;1-{\mathrm{e}}^{-2\xi }),$$ $\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\mu$: general order, $\nu$: general degree and $\xi >0$: variable

Referenced by:

§14.19(ii), Erratum (V1.0.1) for Equation (14.19.2)

Permalink:

http://dlmf.nist.gov/14.19.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.1):

Originally the argument to $𝐅$ was incorrect and the condition on $\mu$ too weak:

$$P_{\nu -\frac{1}{2}}^{\mu }\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(1-2\mu \right)2^{2\mu }}{\mathrm{\Gamma }\left(1-\mu \right){\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}𝐅(\frac{1}{2}-\mu ,\frac{1}{2}+\nu -\mu ;1-2\mu ;{\mathrm{e}}^{-2\xi }),$$ $\mu \ne \frac{1}{2}$.

Also, the factor multiplying $𝐅$ was rewritten to clarify the poles, which determine the correct condition on $\mu$.

Reported 2010-11-02 by Alvaro Valenzuela

See also:

Annotations for §14.19(ii), §14.19 and Ch.14

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37: 10.68 Modulus and Phase Functions

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${\mathrm{ber}}_{\nu }x=M_{\nu }\left(x\right)\cos {\theta }_{\nu }\left(x\right),$

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${\mathrm{bei}}_{\nu }x=M_{\nu }\left(x\right)\sin {\theta }_{\nu }\left(x\right),$

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${\ker }_{\nu }x=N_{\nu }\left(x\right)\cos {\varphi }_{\nu }\left(x\right),$

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§10.68(iii) Asymptotic Expansions for Large Argument

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38: 7.1 Special Notation

… ►Unless otherwise noted, primes indicate derivatives with respect to the argument. ►The main functions treated in this chapter are the error function $\mathrm{erf}z$; the complementary error functions $\mathrm{erfc}z$ and $w\left(z\right)$; Dawson’s integral $F\left(z\right)$; the Fresnel integrals $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$; the Goodwin–Staton integral $G\left(z\right)$; the repeated integrals of the complementary error function ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$; the Voigt functions $𝖴(x,t)$ and $𝖵(x,t)$. ►Alternative notations are $Q(z)=\frac{1}{2}\mathrm{erfc}\left(z/\sqrt{2}\right)$, $P(z)=\mathrm{\Phi }(z)=\frac{1}{2}\mathrm{erfc}\left(-z/\sqrt{2}\right)$, $\mathrm{Erf}z=\frac{1}{2}\sqrt{\pi }\mathrm{erf}z$, $\mathrm{Erfi}z={\mathrm{e}}^{z^2}F\left(z\right)$, $C_1(z)=C\left(\sqrt{2/\pi }z\right)$, $S_1(z)=S\left(\sqrt{2/\pi }z\right)$, $C_2(z)=C\left(\sqrt{2z/\pi }\right)$, $S_2(z)=S\left(\sqrt{2z/\pi }\right)$. …

39: Bibliography F

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Theory and Computation of Spheroidal Harmonics with General Arguments. Master’s Thesis, The University of Western Australia, Department of Physics.

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Notes:

Contains Mathematica package for spheroidal wave functions of complex arguments with complex parameters.

External Links:

FullText

Referenced by:

§30.12, 1st item, 2nd item.

Encodings:

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Complex roots of $\sin z=az$, $\cos z=az$, and $\cosh z=az$ . Math. Comp. 30 (135), pp. 541–545.

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External Links:

FullText, MathReview (Gerhard Merz), ZentralBlatt

Referenced by:

§4.46.

Encodings:

BibTeX

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Tables of Weber Parabolic Cylinder Functions and Other Functions for Large Arguments. National Physical Laboratory Mathematical Tables, Vol. 4. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.

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External Links:

MathReview (L. J. Slater), ZentralBlatt

Referenced by:

3rd item, 2nd item.

Encodings:

BibTeX

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Error bounds for the asymptotic expansion of the ratio of two gamma functions with complex argument. SIAM J. Math. Anal. 23 (2), pp. 505–511.

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External Links:

ISSN 0036-1410, Document, FullText, MathReview (A. Reinfelds), ZentralBlatt

Referenced by:

§5.11(iii), §5.11(iii).

Encodings:

BibTeX

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Fast computation of incomplete elliptic integral of first kind by half argument transformation. Numer. Math. 116 (4), pp. 687–719.

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External Links:

ISSN 0029-599X, Document, FullText, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

§19.36(iv), §19.36(iv).

Encodings:

BibTeX

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40: 10.61 Definitions and Basic Properties

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§10.61(iii) Reflection Formulas for Arguments

►In general, Kelvin functions have a branch point at $x=0$ and functions with arguments $x{\mathrm{e}}^{\pm \pi \mathrm{i}}$ are complex. … ►

§10.61(iv) Reflection Formulas for Orders

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$${\ker }_{-\nu }x=\cos \left(\nu \pi \right){\ker }_{\nu }x-\sin \left(\nu \pi \right){\mathrm{kei}}_{\nu }x,$$

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$${\mathrm{kei}}_{-\nu }x=\sin \left(\nu \pi \right){\ker }_{\nu }x+\cos \left(\nu \pi \right){\mathrm{kei}}_{\nu }x.$$

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