series of cosecants or cotangents

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11: 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:

pMML, png, TeX

See also:

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:

pMML, png, TeX

See also:

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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

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

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12: 4.42 Solution of Triangles

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4.42.3 $$\tan A=\frac{a}{b}=\frac{1}{\cot A}.$$

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Symbols:

$\cot z$: cotangent function, $\tan z$: tangent function, $A$: angle, $a$: height and $b$: base

A&S Ref:

4.3.147

Permalink:

http://dlmf.nist.gov/4.42.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(i), §4.42 and Ch.4

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4.42.11 $$\cos a\cos C=\sin a\cot b-\sin C\cot B,$$

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Symbols:

$\cos z$: cosine function, $\cot z$: cotangent function, $\sin z$: sine function, $B$: angle, $C$: angle, $a$: arc length and $b$: arc length

Permalink:

http://dlmf.nist.gov/4.42.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

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13: 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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14: 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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$\coth z$	$\mathrm{\infty }$	$0$	$\mathrm{\infty }$	$0$	$1$

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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: 14.11 Derivatives with Respect to Degree or Order

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14.11.1 $$\frac{\partial }{\partial \nu }{𝖯}_{\nu }^{\mu }\left(x\right)=\pi \cot \left(\nu \pi \right){𝖯}_{\nu }^{\mu }\left(x\right)-\frac{1}{\pi }{𝖠}_{\nu }^{\mu }(x),$$

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Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $\mu$: general order, $\nu$: general degree and ${𝖠}_{\nu }^{\mu }(x)$

Referenced by:

§14.11, §14.11

Permalink:

http://dlmf.nist.gov/14.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

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14.11.2 $$\frac{\partial }{\partial \nu }{𝖰}_{\nu }^{\mu }\left(x\right)=-\frac{1}{2}{\pi }^2{𝖯}_{\nu }^{\mu }\left(x\right)+\frac{\pi \sin \left(\mu \pi \right)}{\sin \left(\nu \pi \right)\sin \left((\nu +\mu )\pi \right)}{𝖰}_{\nu }^{\mu }\left(x\right)-\frac{1}{2}\cot \left((\nu +\mu )\pi \right){𝖠}_{\nu }^{\mu }(x)+\frac{1}{2}\csc \left((\nu +\mu )\pi \right){𝖠}_{\nu }^{\mu }(-x),$$

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Symbols:

: Ferrers function of the first kind, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $x$: real variable, $\mu$: general order, $\nu$: general degree and ${𝖠}_{\nu }^{\mu }(x)$

Referenced by:

§14.11

Permalink:

http://dlmf.nist.gov/14.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

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14.11.4 $${\frac{\partial }{\partial \mu }{𝖯}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}=\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){𝖯}_{\nu }\left(x\right)+{𝖰}_{\nu }\left(x\right),$$

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Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.11, §14.11

Permalink:

http://dlmf.nist.gov/14.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

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14.11.5 $${\frac{\partial }{\partial \mu }{𝖰}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}=-\frac{1}{4}{\pi }^2{𝖯}_{\nu }\left(x\right)+\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){𝖰}_{\nu }\left(x\right).$$

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Symbols:

: Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.11

Permalink:

http://dlmf.nist.gov/14.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

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17: 4.20 Derivatives and Differential Equations

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4.20.4 $$\frac{d}{dz}\csc z=-\csc z\cot z,$$

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Symbols:

$\csc z$: cosecant function, $\cot z$: cotangent function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: complex variable

A&S Ref:

4.3.108

Permalink:

http://dlmf.nist.gov/4.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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4.20.6 $$\frac{d}{dz}\cot z=-{\csc }^{2}z,$$

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Symbols:

$\csc z$: cosecant function, $\cot z$: cotangent function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: complex variable

A&S Ref:

4.3.110

Permalink:

http://dlmf.nist.gov/4.20.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.20 and Ch.4

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18: 4.47 Approximations

… ►Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arctan$, $\mathrm{arcsinh}$. … ►Hart et al. (1968) give $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arccos$, $\arctan$, $\sinh$, $\cosh$, $\tanh$, $\mathrm{arcsinh}$, $\mathrm{arccosh}$. …

19: 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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

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

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20: 4.30 Elementary Properties

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Table 4.30.1: Hyperbolic functions: interrelations. …

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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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