cotangent function

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11: 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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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. …

12: 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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13: 4.21 Identities

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4.21.5 $$\cot \left(u\pm v\right)=\frac{\pm \cot u\cot v-1}{\cot u\pm \cot v}.$$

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Symbols:

$\cot z$: cotangent function

A&S Ref:

4.3.19

Permalink:

http://dlmf.nist.gov/4.21.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(i), §4.21 and Ch.4

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4.21.11 $$\cot u\pm \cot v=\frac{\sin \left(v\pm u\right)}{\sin u\sin v}.$$

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Symbols:

$\cot z$: cotangent function and $\sin z$: sine function

A&S Ref:

4.3.39

Permalink:

http://dlmf.nist.gov/4.21.E11

Encodings:

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Annotations for §4.21(i), §4.21 and Ch.4

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4.21.14 $${\csc }^{2}z=1+{\cot }^{2}z.$$

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Symbols:

$\csc z$: cosecant function, $\cot z$: cotangent function and $z$: complex variable

A&S Ref:

4.3.12

Permalink:

http://dlmf.nist.gov/4.21.E14

Encodings:

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Annotations for §4.21(ii), §4.21 and Ch.4

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4.21.29 $$\tan \left(2z\right)=\frac{2\tan z}{1-{\tan }^{2}z}=\frac{2\cot z}{{\cot }^{2}z-1}=\frac{2}{\cot z-\tan z}.$$

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Symbols:

$\cot z$: cotangent function, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.26

Permalink:

http://dlmf.nist.gov/4.21.E29

Encodings:

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See also:

Annotations for §4.21(iii), §4.21 and Ch.4

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4.21.40 $$\cot z=\frac{\sin \left(2x\right)-\mathrm{i}\sinh \left(2y\right)}{\cosh \left(2y\right)-\cos \left(2x\right)}.$$

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Symbols:

$\cos z$: cosine function, $\cot z$: cotangent function, $\cosh z$: hyperbolic cosine 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:

4.3.58

Permalink:

http://dlmf.nist.gov/4.21.E40

Encodings:

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See also:

Annotations for §4.21(iv), §4.21 and Ch.4

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14: 4.22 Infinite Products and Partial Fractions

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4.22.3 $$\cot z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{1}{z^2-n^2{\pi }^2},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.91

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/4.22.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

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15: 4.36 Infinite Products and Partial Fractions

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4.36.3 $$\coth z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{1}{z^2+n^2{\pi }^2},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\coth z$: hyperbolic cotangent function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.36.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.36 and Ch.4

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16: 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

Encodings:

pMML, png, TeX

See also:

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

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17: 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

Encodings:

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See also:

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

Encodings:

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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

Encodings:

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See also:

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

Encodings:

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See also:

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

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18: 5.4 Special Values and Extrema

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5.4.16 $$\mathrm{\Im }\psi \left(\mathrm{i}y\right)=\frac{1}{2y}+\frac{\pi }{2}\coth \left(\pi y\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\coth z$: hyperbolic cotangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.3.11

Permalink:

http://dlmf.nist.gov/5.4.E16

Encodings:

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See also:

Annotations for §5.4(ii), §5.4 and Ch.5

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5.4.18 $$\mathrm{\Im }\psi \left(1+\mathrm{i}y\right)=-\frac{1}{2y}+\frac{\pi }{2}\coth \left(\pi y\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\coth z$: hyperbolic cotangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.3.13

Referenced by:

§5.4(ii)

Permalink:

http://dlmf.nist.gov/5.4.E18

Encodings:

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See also:

Annotations for §5.4(ii), §5.4 and Ch.5

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5.4.19 $$\psi \left(\frac{p}{q}\right)=-\gamma -\ln q-\frac{\pi }{2}\cot \left(\frac{\pi p}{q}\right)+\frac{1}{2}\sum _{k=1}^{q-1}\cos \left(\frac{2\pi kp}{q}\right)\ln \left(2-2\cos \left(\frac{2\pi k}{q}\right)\right).$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function, $q$: real or complex variable and $k$: nonnegative integer

Referenced by:

§5.19(i), §5.4(ii)

Permalink:

http://dlmf.nist.gov/5.4.E19

Encodings:

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See also:

Annotations for §5.4(ii), §5.4 and Ch.5

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19: 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

Encodings:

pMML, png, TeX

See also:

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

Encodings:

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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:

pMML, png, TeX

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). …

20: 4.15 Graphics

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4.15.2 $$\cot \left(x+\mathrm{i}y\right)=-\tan \left(x+\frac{1}{2}\pi +\mathrm{i}y\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\mathrm{i}$: imaginary unit, $\tan z$: tangent function, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.15(iii), §4.15 and Ch.4

… ►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).