hyperbolic cotangent 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: 4.31 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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3: 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

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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

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Annotations for §4.34 and Ch.4

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4.34.14 $$w=(1/a)\coth \left(az+c\right),$$

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Symbols:

$\coth z$: hyperbolic cotangent function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E14

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Annotations for §4.34 and Ch.4

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4: 4.28 Definitions and Periodicity

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4.28.7 $$\coth z=\frac{1}{\tanh z}.$$

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Defines:

$\coth z$: hyperbolic cotangent function

Symbols:

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

A&S Ref:

4.5.6

Referenced by:

§4.45(i), §4.45(ii)

Permalink:

http://dlmf.nist.gov/4.28.E7

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Annotations for §4.28 and Ch.4

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4.28.13 $$\cot \left(\mathrm{i}z\right)=-\mathrm{i}\coth z.$$

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Symbols:

$\cot z$: cotangent function, $\coth z$: hyperbolic cotangent function, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.5.12

Referenced by:

§4.29(ii), §4.33

Permalink:

http://dlmf.nist.gov/4.28.E13

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Annotations for §4.28, §4.28 and Ch.4

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5: 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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$\coth \theta$	$a^{-1}{(a^2+1)}^{1/2}$	$a{(a^2-1)}^{-1/2}$	$a^{-1}$	${(1+a^2)}^{1/2}$	${(1-a^2)}^{-1/2}$	$a$

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6: 4.29 Graphics

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§4.29(i) Real Arguments

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Figure 4.29.3: $\tanh x$ and $\coth x$. 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. …

7: 13.24 Series

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13.24.3 $$\exp \left(-\frac{1}{2}z\left(\coth t-\frac{1}{t}\right)\right){\left(\frac{t}{\sinh t}\right)}^{1-2\mu }=\sum _{s=0}^{\mathrm{\infty }}{p}_s^{(\mu )}(z){\left(-\frac{t}{z}\right)}^s.$$

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Symbols:

$\exp z$: exponential function, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $s$: nonnegative integer, $z$: complex variable and $p_s$: coefficents

Permalink:

http://dlmf.nist.gov/13.24.E3

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Annotations for §13.24(ii), §13.24 and Ch.13

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8: 4.35 Identities

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4.35.4 $$\coth \left(u\pm v\right)=\frac{\pm \coth u\coth v+1}{\coth u\pm \coth v}.$$

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Symbols:

$\coth z$: hyperbolic cotangent function

A&S Ref:

4.5.27 (modified)

Permalink:

http://dlmf.nist.gov/4.35.E4

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Annotations for §4.35(i), §4.35 and Ch.4

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4.35.10 $$\coth u\pm \coth v=\frac{\sinh \left(v\pm u\right)}{\sinh u\sinh v}.$$

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Symbols:

$\coth z$: hyperbolic cotangent function and $\sinh z$: hyperbolic sine function

A&S Ref:

4.5.46 (modified)

Permalink:

http://dlmf.nist.gov/4.35.E10

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Annotations for §4.35(i), §4.35 and Ch.4

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4.35.13 $${\mathrm{csch}}^{2}z={\coth }^{2}z-1.$$

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Symbols:

$\mathrm{csch}z$: hyperbolic cosecant function, $\coth z$: hyperbolic cotangent function and $z$: complex variable

A&S Ref:

4.5.18

Permalink:

http://dlmf.nist.gov/4.35.E13

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Annotations for §4.35(ii), §4.35 and Ch.4

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4.35.37 $$\coth z=\frac{\sinh \left(2x\right)-\mathrm{i}\sin \left(2y\right)}{\cosh \left(2x\right)-\cos \left(2y\right)}.$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.5.52

Permalink:

http://dlmf.nist.gov/4.35.E37

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Annotations for §4.35(iv), §4.35 and Ch.4

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9: 14.19 Toroidal (or Ring) Functions

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14.19.7 $$P_{n-\frac{1}{2}}^m\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(n+m+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(n-m+\frac{1}{2}\right)}{\left(\frac{2}{\pi \sinh \xi }\right)}^{1/2}{𝑸}_{m-\frac{1}{2}}^n\left(\coth \xi \right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $m$: nonnegative integer, $n$: nonnegative integer and $\xi >0$: variable

Permalink:

http://dlmf.nist.gov/14.19.E7

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Annotations for §14.19(v), §14.19 and Ch.14

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14.19.8 $${𝑸}_{n-\frac{1}{2}}^m\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(m-n+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(m+n+\frac{1}{2}\right)}{\left(\frac{\pi }{2\sinh \xi }\right)}^{1/2}{P}_{m-\frac{1}{2}}^n\left(\coth \xi \right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $m$: nonnegative integer, $n$: nonnegative integer and $\xi >0$: variable

Permalink:

http://dlmf.nist.gov/14.19.E8

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Annotations for §14.19(v), §14.19 and Ch.14

10: 4.37 Inverse Hyperbolic Functions

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4.37.6 $$\mathrm{Arccoth}z=\mathrm{Arctanh}\left(1/z\right).$$

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Defines:

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

Symbols:

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

Permalink:

http://dlmf.nist.gov/4.37.E6

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Annotations for §4.37(i), §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

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

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4.37.15 $$\mathrm{arccoth}\left(-z\right)=-\mathrm{arccoth}z,$$ $z\ne \pm 1$.

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Symbols:

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

Permalink:

http://dlmf.nist.gov/4.37.E15

Encodings:

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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). …