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1: 4.37 Inverse Hyperbolic Functions

§4.37 Inverse Hyperbolic Functions

… ►The principal values of the inverse hyperbolic cosecant, hyperbolic secant, and hyperbolic tangent are given by … ►

Inverse Hyperbolic Sine

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Inverse Hyperbolic Cosine

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Inverse Hyperbolic Tangent

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2: 36.2 Catastrophes and Canonical Integrals

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Normal Forms for Umbilic Catastrophes with Codimension $K=3$

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36.2.3 $${\mathrm{\Phi }}^{(\mathrm{H})}(s,t;𝐱)=s^3+t^3+zst+yt+xs,$$ $𝐱=\{x,y,z\}$,

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Defines:

${\mathrm{\Phi }}^{(\mathrm{H})}(s,t;𝐱)$: hyperbolic umbilic catastrophe

Symbols:

$y$: real parameter, $z$: real parameter, $t$: variable, $s$: variable and $x$: real parameter

Referenced by:

§36.10(iii), §36.2(i), §36.8

Permalink:

http://dlmf.nist.gov/36.2.E3

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Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

►(hyperbolic umbilic). ►

Canonical Integrals

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36.2.27 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,z\right)={\mathrm{\Psi }}^{(\mathrm{H})}\left(y,x,z\right).$$

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Symbols:

${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.2.E27

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See also:

Annotations for §36.2(iii), §36.2 and Ch.36

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3: 4.35 Identities

§4.35 Identities

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§4.35(i) Addition Formulas

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§4.35(ii) Squares and Products

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§4.35(iii) Multiples of the Argument

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§4.35(iv) Real and Imaginary Parts; Moduli

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4: 4.29 Graphics

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§4.29(i) Real Arguments

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Figure 4.29.6: Principal values of $\mathrm{arccsch}x$ and $\mathrm{arcsech}x$. … Magnify

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§4.29(ii) Complex Arguments

►The conformal mapping $w=\sinh z$ is obtainable from Figure 4.15.7 by rotating both the $w$-plane and the $z$-plane through an angle $\frac{1}{2}\pi$, compare (4.28.8). ►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. …

5: 4.28 Definitions and Periodicity

§4.28 Definitions and Periodicity

… ►

Relations to Trigonometric Functions

… ►As a consequence, many properties of the hyperbolic functions follow immediately from the corresponding properties of the trigonometric functions. ►

Periodicity and Zeros

►The functions $\sinh z$ and $\cosh z$ have period $2\pi \mathrm{i}$, and $\tanh z$ has period $\pi \mathrm{i}$. …

6: 4.1 Special Notation

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$k,m,n$	integers.
…

… ►The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$. … ►; 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. …

7: 4.32 Inequalities

§4.32 Inequalities

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4.32.1 $$\cosh x\le {\left(\frac{\sinh x}{x}\right)}^3,$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E1

Encodings:

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See also:

Annotations for §4.32 and Ch.4

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4.32.2 $x>0$,

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Symbols:

$\cos z$: cosine function, $\tanh z$: hyperbolic tangent function, $\sin z$: sine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E2

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Annotations for §4.32 and Ch.4

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4.32.3 $$|\cosh x-\cosh y|\ge |x-y|\sqrt{\sinh x\sinh y},$$ $x>0$, $y>0$,

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Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E3

Encodings:

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Annotations for §4.32 and Ch.4

… ►For these and other inequalities involving hyperbolic functions see Mitrinović (1964, pp. 61, 76, 159) and Mitrinović (1970, p. 270).

8: 4.31 Special Values and Limits

§4.31 Special Values and Limits

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Table 4.31.1: Hyperbolic functions: values at multiples of $\frac{1}{2}\pi \mathrm{i}$.

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$z$	0	$\frac{1}{2}\pi \mathrm{i}$	$\pi \mathrm{i}$	$\frac{3}{2}\pi \mathrm{i}$	$\mathrm{\infty }$
…

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4.31.1 $$\underset{z\to 0}{lim}\frac{\sinh z}{z}=1,$$

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Symbols:

$\sinh z$: hyperbolic sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E1

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Annotations for §4.31 and Ch.4

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4.31.2 $$\underset{z\to 0}{lim}\frac{\tanh z}{z}=1,$$

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Symbols:

$\tanh z$: hyperbolic tangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E2

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Annotations for §4.31 and Ch.4

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4.31.3 $$\underset{z\to 0}{lim}\frac{\cosh z-1}{z^2}=\frac{1}{2}.$$

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Symbols:

$\cosh z$: hyperbolic cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E3

Encodings:

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See also:

Annotations for §4.31 and Ch.4

9: 4.41 Sums

§4.41 Sums

►For sums of hyperbolic functions see Gradshteyn and Ryzhik (2015, Chapter 1), Hansen (1975, §43), Prudnikov et al. (1986a, §5.3), and Zucker (1979).

10: 4.34 Derivatives and Differential Equations

§4.34 Derivatives and Differential Equations

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4.34.4 $$\frac{d}{dz}\mathrm{csch}z=-\mathrm{csch}z\coth z,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{csch}z$: hyperbolic cosecant function, $\coth z$: hyperbolic cotangent function and $z$: complex variable

A&S Ref:

4.5.74

Permalink:

http://dlmf.nist.gov/4.34.E4

Encodings:

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Annotations for §4.34 and Ch.4

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4.34.5 $$\frac{d}{dz}\mathrm{sech}z=-\mathrm{sech}z\tanh z,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{sech}z$: hyperbolic secant function, $\tanh z$: hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.5.75

Permalink:

http://dlmf.nist.gov/4.34.E5

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Annotations for §4.34 and Ch.4

… ►With $a\ne 0$, the general solutions of the differential equations … ►

4.34.11 $$w=A\cosh \left(az\right)+B\sinh \left(az\right),$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $a$: real or complex constant, $z$: complex variable, $w$: solution, $A$: arbitrary constant and $B$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E11

Encodings:

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See also:

Annotations for §4.34 and Ch.4

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