inverse trigonometric functions

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1: 4.23 Inverse Trigonometric Functions

§4.23 Inverse Trigonometric Functions

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§4.23(i) General Definitions

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Figure 4.23.1: $z$-plane. …

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§4.23(iv) Logarithmic Forms

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§4.23(vii) Special Values and Interrelations

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2: 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).

3: 4.1 Special Notation

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$k,m,n$	integers.
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… ►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. …

4: 19.10 Relations to Other Functions

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19.10.2 $$(\sinh \varphi )R_C(1,{\cosh }^2\varphi )=\mathrm{gd}\left(\varphi \right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{gd}x$: Gudermannian function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $\varphi$: real or complex argument

Referenced by:

§19.10(ii), §19.6(ii)

Permalink:

http://dlmf.nist.gov/19.10.E2

Encodings:

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See also:

Annotations for §19.10(ii), §19.10 and Ch.19

5: 4.29 Graphics

… ►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. …

6: 4.46 Tables

§4.46 Tables

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7: 4.47 Approximations

§4.47 Approximations

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§4.47(i) Chebyshev-Series Expansions

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8: 4.37 Inverse Hyperbolic Functions

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4.37.4 $$\mathrm{Arccsch}z=\mathrm{Arcsinh}\left(1/z\right),$$

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Defines:

$\mathrm{Arccsch}z$: general inverse hyperbolic cosecant function

Symbols:

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

Permalink:

http://dlmf.nist.gov/4.37.E4

Encodings:

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See also:

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

… ►Each is two-valued on the corresponding cut(s), and each is real on the part of the real axis that remains after deleting the intersections with the corresponding cuts. … ►

4.37.7 $$\mathrm{arccsch}z=\mathrm{arcsinh}\left(1/z\right),$$

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Defines:

$\mathrm{arccsch}z$: inverse hyperbolic cosecant function

Symbols:

$\mathrm{arcsinh}z$: inverse hyperbolic sine function and $z$: complex variable

A&S Ref:

4.6.4

Referenced by:

§4.37(iv), §4.45(i), §4.45(ii)

Permalink:

http://dlmf.nist.gov/4.37.E7

Encodings:

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See also:

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

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4.37.8 $$\mathrm{arcsech}z=\mathrm{arccosh}\left(1/z\right).$$

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Defines:

$\mathrm{arcsech}z$: inverse hyperbolic secant function

Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function and $z$: complex variable

A&S Ref:

4.6.5

Permalink:

http://dlmf.nist.gov/4.37.E8

Encodings:

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See also:

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

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4.37.9 $$\mathrm{arccoth}z=\mathrm{arctanh}\left(1/z\right),$$ $z\ne \pm 1$.

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Defines:

$\mathrm{arccoth}z$: inverse hyperbolic cotangent function

Symbols:

$\mathrm{arctanh}z$: inverse hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.6.6

Referenced by:

§4.37(iv), §4.45(i), §4.45(ii)

Permalink:

http://dlmf.nist.gov/4.37.E9

Encodings:

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See also:

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

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9: 4.15 Graphics

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§4.15(i) Real Arguments

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Figure 4.15.7: Conformal mapping of sine and inverse sine. … Magnify

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§4.15(iii) Complex Arguments: Surfaces

… ►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).

10: 4.32 Inequalities

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4.32.4 $$\arctan x\le \frac{1}{2}\pi \tanh x,$$ $x\ge 0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\tanh z$: hyperbolic tangent function, $\arctan z$: arctangent function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E4

Encodings:

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See also:

Annotations for §4.32 and Ch.4

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