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1: 4.37 Inverse Hyperbolic Functions

§4.37 Inverse Hyperbolic Functions

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Inverse Hyperbolic Sine

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Inverse Hyperbolic Cosine

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Inverse Hyperbolic Tangent

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Other Inverse Functions

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2: 4.23 Inverse Trigonometric Functions

§4.23 Inverse Trigonometric Functions

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Inverse Sine

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Inverse Cosine

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Inverse Tangent

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Other Inverse Functions

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3: 22.15 Inverse Functions

§22.15 Inverse Functions

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§22.15(i) Definitions

… ►Each of these inverse functions is multivalued. The principal values satisfy … ►

4: 4.27 Sums

§4.27 Sums

►For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2015, Chapter 1), Hansen (1975, §§14–42), Oberhettinger (1973), and Prudnikov et al. (1986a, Chapter 5).

5: 4.47 Approximations

§4.47 Approximations

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§4.47(i) Chebyshev-Series Expansions

►Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arctan$, $\mathrm{arcsinh}$. … ►Hart et al. (1968) give $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arccos$, $\arctan$, $\sinh$, $\cosh$, $\tanh$, $\mathrm{arcsinh}$, $\mathrm{arccosh}$. … ►Luke (1975, Chapter 3) supplies real and complex approximations for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\arctan$, $\mathrm{arcsinh}$. …

6: 4.29 Graphics

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§4.29(i) Real Arguments

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Figure 4.29.2: Principal values of $\mathrm{arcsinh}x$ and $\mathrm{arccosh}x$. … Magnify

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Figure 4.29.4: Principal values of $\mathrm{arctanh}x$ and $\mathrm{arccoth}x$. … Magnify

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§4.29(ii) Complex Arguments

… ►The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …

7: 4.1 Special Notation

… ►The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$. ►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. ►Sometimes in the literature the meanings of $\ln$ and $\mathrm{Ln}$ are interchanged; similarly for $\arcsin z$ and $\mathrm{Arcsin}z$, etc. …${\sin }^{-1}z$ for $\arcsin z$ and ${\mathrm{Sin}}^{-1}z$ for $\mathrm{Arcsin}z$.

8: 4.24 Inverse Trigonometric Functions: Further Properties

§4.24 Inverse Trigonometric Functions: Further Properties

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§4.24(i) Power Series

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§4.24(ii) Derivatives

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§4.24(iii) Addition Formulas

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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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9: 7.17 Inverse Error Functions

§7.17 Inverse Error Functions

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§7.17(i) Notation

►The inverses of the functions $x=\mathrm{erf}y$, $x=\mathrm{erfc}y$, $y\in ℝ$, are denoted by … ►

§7.17(ii) Power Series

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§7.17(iii) Asymptotic Expansion of $\mathrm{inverfc}x$ for Small $x$

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10: 4.38 Inverse Hyperbolic Functions: Further Properties

§4.38 Inverse Hyperbolic Functions: Further Properties

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§4.38(i) Power Series

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§4.38(ii) Derivatives

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§4.38(iii) Addition Formulas

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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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