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1: 4.24 Inverse Trigonometric Functions: Further Properties

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4.24.14 $$\mathrm{Arccos}u\pm \mathrm{Arccos}v=\mathrm{Arccos}\left(uv\mp {((1-u^2)(1-v^2))}^{1/2}\right),$$

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Symbols:

$\mathrm{Arccos}z$: general arccosine function

A&S Ref:

4.4.33

Permalink:

http://dlmf.nist.gov/4.24.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

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4.24.16 $$\mathrm{Arcsin}u\pm \mathrm{Arccos}v=\mathrm{Arcsin}\left(uv\pm {((1-u^2)(1-v^2))}^{1/2}\right)=\mathrm{Arccos}\left(v{(1-u^2)}^{1/2}\mp u{(1-v^2)}^{1/2}\right),$$

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Symbols:

$\mathrm{Arccos}z$: general arccosine function and $\mathrm{Arcsin}z$: general arcsine function

A&S Ref:

4.4.35

Permalink:

http://dlmf.nist.gov/4.24.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

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2: 4.38 Inverse Hyperbolic Functions: Further Properties

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4.38.10 $$\frac{d}{dz}\mathrm{arccosh}z=\pm {(z^2-1)}^{-1/2},$$ $\mathrm{\Re }z\gtrless 0$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.6.38 (has an error.)

Permalink:

http://dlmf.nist.gov/4.38.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(ii), §4.38 and Ch.4

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4.38.16 $$\mathrm{Arccosh}u\pm \mathrm{Arccosh}v=\mathrm{Arccosh}\left(uv\pm {((u^2-1)(v^2-1))}^{1/2}\right),$$

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Symbols:

$\mathrm{Arccosh}z$: general inverse hyperbolic cosine function

A&S Ref:

4.6.27

Permalink:

http://dlmf.nist.gov/4.38.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(iii), §4.38 and Ch.4

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4.38.18 $$\mathrm{Arcsinh}u\pm \mathrm{Arccosh}v=\mathrm{Arcsinh}\left(uv\pm {((1+u^2)(v^2-1))}^{1/2}\right)=\mathrm{Arccosh}\left(v{(1+u^2)}^{1/2}\pm u{(v^2-1)}^{1/2}\right),$$

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Symbols:

$\mathrm{Arccosh}z$: general inverse hyperbolic cosine function and $\mathrm{Arcsinh}z$: general inverse hyperbolic sine function

A&S Ref:

4.6.29

Permalink:

http://dlmf.nist.gov/4.38.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(iii), §4.38 and Ch.4

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3: 4.37 Inverse Hyperbolic Functions

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4.37.11 $$\mathrm{arccosh}\left(-z\right)=\pm \pi \mathrm{i}+\mathrm{arccosh}z,$$ $\mathrm{\Im }z\gtrless 0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.6.12 (has an error even in the tenth printing.)

Referenced by:

§4.37(iii)

Permalink:

http://dlmf.nist.gov/4.37.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iii), §4.37 and Ch.4

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4.37.19 $$\mathrm{arccosh}z=\ln \left(\pm {(z^2-1)}^{1/2}+z\right),$$ $z\in ℂ\setminus (-\mathrm{\infty },1)$,

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Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $\setminus$: set subtraction and $z$: complex variable

Referenced by:

§4.37(iv), §4.37(iii), §4.37(iv)

Permalink:

http://dlmf.nist.gov/4.37.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

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4.37.22 $$\mathrm{arccosh}x=\pm \ln \left(\mathrm{i}{(1-x^2)}^{1/2}+x\right),$$ $x\in (-1,1]$,

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Symbols:

$\in$: element of, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $(a,b]$: half-closed interval and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.37.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

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4.37.23 $$\mathrm{arccosh}x=\pm \pi \mathrm{i}+\ln \left({(x^2-1)}^{1/2}-x\right),$$ $x\in (-\mathrm{\infty },-1]$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $(a,b]$: half-closed interval and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.37.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

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4.37.30 $$w=\mathrm{Arccosh}z=\pm \mathrm{arccosh}z+2k\pi \mathrm{i},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\mathrm{Arccosh}z$: general inverse hyperbolic cosine function, $k$: integer and $z$: complex variable

A&S Ref:

4.6.9

Permalink:

http://dlmf.nist.gov/4.37.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(v), §4.37 and Ch.4

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4: 4.23 Inverse Trigonometric Functions

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4.23.24 $$\arccos x=\mp \mathrm{i}\ln \left({(x^2-1)}^{1/2}+x\right),$$ $x\in [1,\mathrm{\infty })$,

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Symbols:

$[a,b)$: half-closed interval, $\in$: element of, $\mathrm{i}$: imaginary unit, $\arccos z$: arccosine function, $\ln z$: principal branch of logarithm function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.23.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

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4.23.25 $$\arccos x=\pi \mp \mathrm{i}\ln \left({(x^2-1)}^{1/2}-x\right),$$ $x\in (-\mathrm{\infty },-1]$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{i}$: imaginary unit, $\arccos z$: arccosine function, $\ln z$: principal branch of logarithm function, $(a,b]$: half-closed interval and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.23.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

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4.23.32 $$w=\mathrm{Arccos}z=\pm \arccos z+2k\pi ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\arccos z$: arccosine function, $\mathrm{Arccos}z$: general arccosine function, $k$: integer and $z$: complex variable

A&S Ref:

4.4.11

Permalink:

http://dlmf.nist.gov/4.23.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(v), §4.23 and Ch.4

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5: Bibliography C

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An extension of the method of steepest descents. Proc. Cambridge Philos. Soc. 53, pp. 599–611.

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External Links:

MathReview (P. Henrici), ZentralBlatt

Referenced by:

§2.4(v).

Encodings:

BibTeX

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Advanced Combinatorics: The Art of Finite and Infinite Expansions. enlarged edition, D. Reidel Publishing Co., Dordrecht.

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Notes:

Revised and enlarged English translation by D. Reidel of Analyse Combinatoire, Tomes I, II, Presses Universitaires de France, Paris, 1970.

External Links:

ISBN 90-277-0441-4, MathReview, ZentralBlatt

Referenced by:

§26.3(i), §26.3(iii), §26.3(iv), §26.3(v), §26.4(i), §26.4(ii), §26.4(iii), §26.5(i), §26.5(ii), §26.5(iii), §26.6(i), §26.6(ii), §26.6(iii), §26.7(i), §26.7(ii), §26.7(iii), §26.8(i), §26.8(ii), §26.8(iv), §26.8(v), Ch.26.

Encodings:

BibTeX

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6: 2.4 Contour Integrals

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§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method

… ►For examples, proofs, and extensions see Olver (1997b, Chapter 9), Wong (1989, Chapter 7), Olde Daalhuis and Temme (1994), Chester et al. (1957), Bleistein and Handelsman (1975, Chapter 9), and Temme (2015). …

7: Wadim Zudilin

… ►He has been an Associate Professor at Moscow State University (Russia), an Ostrowski Fellow at Institut Henri Poincaré and Université Paris 6 (France), a Humboldt Fellow at the Cologne University (Germany) and a Visiting Researcher at the Max Planck Institute for Mathematics in Bonn (Germany). …

8: Publications

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From B-Spline Mesh Generation to Effective Visualizations for the NIST Digital Library of Mathematical Functions, in Curve and Surface Design, Proceedings of the Sixth International Conference on Curves and Surfaces, Avignon, France June 29–July 5, 2006, pp. 235–243.

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9: Bibliography M

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Relativistic hydrogen atom revisited. Amer. J. Phys. 68, pp. 1050–1055.

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Referenced by:

§18.39(ii).

Encodings:

BibTeX

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10: 18.39 Applications in the Physical Sciences

… ►See Martinez-y-Romero (2000). …