hyperbolic cosecant function

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1: 4.1 Special Notation

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$k,m,n$	integers.
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… ►; 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. …

2: 4.28 Definitions and Periodicity

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4.28.5 $$\mathrm{csch}z=\frac{1}{\sinh z},$$

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Defines:

$\mathrm{csch}z$: hyperbolic cosecant function

Symbols:

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

A&S Ref:

4.5.4

Permalink:

http://dlmf.nist.gov/4.28.E5

Encodings:

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See also:

Annotations for §4.28 and Ch.4

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4.28.11 $$\csc \left(\mathrm{i}z\right)=-\mathrm{i}\mathrm{csch}z,$$

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Symbols:

$\csc z$: cosecant function, $\mathrm{csch}z$: hyperbolic cosecant function, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.5.10

Permalink:

http://dlmf.nist.gov/4.28.E11

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See also:

Annotations for §4.28, §4.28 and Ch.4

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3: 4.29 Graphics

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§4.29(i) Real Arguments

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Figure 4.29.5: $\mathrm{csch}x$ and $\mathrm{sech}x$. 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. …

4: 4.36 Infinite Products and Partial Fractions

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4.36.4 $${\mathrm{csch}}^{2}z=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{{(z-n\pi \mathrm{i})}^2},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{csch}z$: hyperbolic cosecant function, $\mathrm{i}$: imaginary unit, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.36.E4

Encodings:

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Annotations for §4.36 and Ch.4

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4.36.5 $$\mathrm{csch}z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{z^2+n^2{\pi }^2}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{csch}z$: hyperbolic cosecant function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.36.E5

Encodings:

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See also:

Annotations for §4.36 and Ch.4

5: 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

Encodings:

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See also:

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

Encodings:

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See also:

Annotations for §4.31 and Ch.4

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6: 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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See also:

Annotations for §4.34 and Ch.4

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4.34.6 $$\frac{d}{dz}\coth z=-{\mathrm{csch}}^{2}z.$$

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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:

4.5.76

Permalink:

http://dlmf.nist.gov/4.34.E6

Encodings:

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See also:

Annotations for §4.34 and Ch.4

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7: 4.18 Inequalities

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4.18.7 $$|\csc z|\le \mathrm{csch}|y|,$$

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Symbols:

$\csc z$: cosecant function, $\mathrm{csch}z$: hyperbolic cosecant function, $y$: real variable and $z$: complex variable

A&S Ref:

4.3.85

Permalink:

http://dlmf.nist.gov/4.18.E7

Encodings:

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See also:

Annotations for §4.18, §4.18 and Ch.4

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8: 4.30 Elementary Properties

§4.30 Elementary Properties

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Table 4.30.1: Hyperbolic functions: interrelations. All square roots have their principal values when the functions are real, nonnegative, and finite.

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	$\sinh \theta =a$	$\cosh \theta =a$	$\tanh \theta =a$	$\mathrm{csch}\theta =a$	$\mathrm{sech}\theta =a$	$\coth \theta =a$
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$\mathrm{csch}\theta$	$a^{-1}$	${(a^2-1)}^{-1/2}$	$a^{-1}{(1-a^2)}^{1/2}$	$a$	$a{(1-a^2)}^{-1/2}$	${(a^2-1)}^{1/2}$
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9: 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

… ► $\mathrm{Arcsinh}z$ and $\mathrm{Arccsch}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$. … ►

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

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

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4.37.13 $$\mathrm{arccsch}\left(-z\right)=-\mathrm{arccsch}z.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/4.37.E13

Encodings:

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See also:

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

… ►For the corresponding results for $\mathrm{arccsch}z$, $\mathrm{arcsech}z$, and $\mathrm{arccoth}z$, use (4.37.7)–(4.37.9); compare §4.23(iv). …

10: 4.40 Integrals

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4.40.4 $$\int \mathrm{csch}xdx=\ln \left(\tanh \left(\frac{1}{2}x\right)\right),$$ .

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Symbols:

$dx$: differential of $x$ , $\mathrm{csch}z$: hyperbolic cosecant function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.80

Permalink:

http://dlmf.nist.gov/4.40.E4

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Annotations for §4.40(ii), §4.40 and Ch.4

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4.40.14 $$\int \mathrm{arccsch}xdx=x\mathrm{arccsch}x+\mathrm{arcsinh}x,$$ ,

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Symbols:

$dx$: differential of $x$ , $\mathrm{arccsch}z$: inverse hyperbolic cosecant function, $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\int$: integral and $x$: real variable

A&S Ref:

4.6.46 (modified. The first printing had an error.)

Permalink:

http://dlmf.nist.gov/4.40.E14

Encodings:

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See also:

Annotations for §4.40(iv), §4.40 and Ch.4

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