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11: 4.15 Graphics

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§4.15(i) Real Arguments

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Figure 4.15.2: $\mathrm{Arcsin}x$ and $\mathrm{Arccos}x$. … Magnify

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§4.15(iii) Complex Arguments: Surfaces

►In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. … ►The corresponding surfaces for $\arccos \left(x+\mathrm{i}y\right)$, $\mathrm{arccot}\left(x+\mathrm{i}y\right)$, $\mathrm{arcsec}\left(x+\mathrm{i}y\right)$ can be visualized from Figures 4.15.9, 4.15.11, 4.15.13 with the aid of equations (4.23.16)–(4.23.18).

12: 4.24 Inverse Trigonometric Functions: Further Properties

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4.24.13 $$\mathrm{Arcsin}u\pm \mathrm{Arcsin}v=\mathrm{Arcsin}\left(u{(1-v^2)}^{1/2}\pm v{(1-u^2)}^{1/2}\right),$$

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Symbols:

$\mathrm{Arcsin}z$: general arcsine function

A&S Ref:

4.4.32

Permalink:

http://dlmf.nist.gov/4.24.E13

Encodings:

pMML, png, TeX

See also:

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

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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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4.24.15 $$\mathrm{Arctan}u\pm \mathrm{Arctan}v=\mathrm{Arctan}\left(\frac{u\pm v}{1\mp uv}\right),$$

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Symbols:

$\mathrm{Arctan}z$: general arctangent function

A&S Ref:

4.4.34

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.24.E15

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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4.24.17 $$\mathrm{Arctan}u\pm \mathrm{Arccot}v=\mathrm{Arctan}\left(\frac{uv\pm 1}{v\mp u}\right)=\mathrm{Arccot}\left(\frac{v\mp u}{uv\pm 1}\right).$$

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Symbols:

$\mathrm{Arccot}z$: general arccotangent function and $\mathrm{Arctan}z$: general arctangent function

A&S Ref:

4.4.36

Permalink:

http://dlmf.nist.gov/4.24.E17

Encodings:

pMML, png, TeX

See also:

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

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13: 4.1 Special Notation

… ►The main functions treated in this chapter are the logarithm $\ln z$, $\mathrm{Ln}z$; the exponential $\exp z$, ${\mathrm{e}}^z$; the circular trigonometric (or just trigonometric) functions $\sin z$, $\cos z$, $\tan z$, $\csc z$, $\sec z$, $\cot z$; the inverse trigonometric functions $\arcsin z$, $\mathrm{Arcsin}z$, etc. ; the hyperbolic trigonometric (or just hyperbolic) functions $\sinh z$, $\cosh z$, $\tanh z$, $\mathrm{csch}z$, $\mathrm{sech}z$, $\coth z$; the inverse hyperbolic functions $\mathrm{arcsinh}z$, $\mathrm{Arcsinh}z$, etc. …

14: 22.18 Mathematical Applications

… ►This circumvents the cumbersome branch structure of the multivalued functions $x(y)$ or $y(x)$, and constitutes the process of uniformization; see Siegel (1988, Chapter II). …

15: 16.2 Definition and Analytic Properties

… ►Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at $z=0,1$, and $\mathrm{\infty }$. …

16: 15.2 Definitions and Analytical Properties

… ►As a multivalued function of $z$, $𝐅(a,b;c;z)$ is analytic everywhere except for possible branch points at $z=0$, $1$, and $\mathrm{\infty }$. …

17: 4.38 Inverse Hyperbolic Functions: Further Properties

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4.38.15 $$\mathrm{Arcsinh}u\pm \mathrm{Arcsinh}v=\mathrm{Arcsinh}\left(u{(1+v^2)}^{1/2}\pm v{(1+u^2)}^{1/2}\right),$$

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Symbols:

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

A&S Ref:

4.6.26

Permalink:

http://dlmf.nist.gov/4.38.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(iii), §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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4.38.17 $$\mathrm{Arctanh}u\pm \mathrm{Arctanh}v=\mathrm{Arctanh}\left(\frac{u\pm v}{1\pm uv}\right),$$

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Symbols:

$\mathrm{Arctanh}z$: general inverse hyperbolic tangent function

A&S Ref:

4.6.28

Permalink:

http://dlmf.nist.gov/4.38.E17

Encodings:

pMML, png, TeX

See also:

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

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

ⓘ

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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4.38.19 $$\mathrm{Arctanh}u\pm \mathrm{Arccoth}v=\mathrm{Arctanh}\left(\frac{uv\pm 1}{v\pm u}\right)=\mathrm{Arccoth}\left(\frac{v\pm u}{uv\pm 1}\right).$$

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Symbols:

$\mathrm{Arccoth}z$: general inverse hyperbolic cotangent function and $\mathrm{Arctanh}z$: general inverse hyperbolic tangent function

A&S Ref:

4.6.30

Permalink:

http://dlmf.nist.gov/4.38.E19

Encodings:

pMML, png, TeX

See also:

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

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18: 5.9 Integral Representations

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5.9.10 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)=\left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+2{\int }_0^{\mathrm{\infty }}\frac{\arctan \left(t/z\right)}{{\mathrm{e}}^{2\pi t}-1}dt,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

6.1.50

Referenced by:

(5.9.10_1), §5.9(i), §5.9(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)

Permalink:

http://dlmf.nist.gov/5.9.E10

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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5.9.10_1 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)=\left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)-\frac{z}{\pi }{\int }_0^{\mathrm{\infty }}\frac{\ln \left(1-{\mathrm{e}}^{-2\pi t}\right)}{t^2+z^2}dt,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Proof sketch:

Can be obtained from (5.9.10) via integration by parts.

Referenced by:

§5.9(i), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/5.9.E10_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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5.9.10_2 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)=\left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}\left(\frac{1}{{\mathrm{e}}^t-1}-\frac{1}{t}+\frac{1}{2}\right)\frac{dt}{t},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Source:

Whittaker and Watson (1927, §12.31)

Referenced by:

§5.9(i), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/5.9.E10_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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5.9.11 $$\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)=-\gamma z-\frac{1}{2\pi \mathrm{i}}{\int }_{-c-\mathrm{\infty }\mathrm{i}}^{-c+\mathrm{\infty }\mathrm{i}}\frac{\pi z^{-s}}{s\sin \left(\pi s\right)}\zeta \left(-s\right)ds,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $s$: real or complex variable and $z$: complex variable

Referenced by:

§5.9(i), §5.9(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)

Permalink:

http://dlmf.nist.gov/5.9.E11

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z+1\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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19: 10.68 Modulus and Phase Functions

§10.68 Modulus and Phase Functions

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§10.68(i) Definitions

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§10.68(ii) Basic Properties

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§10.68(iii) Asymptotic Expansions for Large Argument

… ►Additional properties of the modulus and phase functions are given in Young and Kirk (1964, pp. xi–xv). …

20: 8.21 Generalized Sine and Cosine Integrals

… ►From §§8.2(i) and 8.2(ii) it follows that each of the four functions $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ is a multivalued function of $z$ with branch point at $z=0$. …