series of cosecants or cotangents

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1: 23.8 Trigonometric Series and Products

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23.8.2 $$\zeta \left(z\right)-\frac{{\eta }_{1}z}{{\omega }_1}-\frac{\pi }{2{\omega }_1}\cot \left(\frac{\pi z}{2{\omega }_1}\right)=\frac{2\pi }{{\omega }_1}\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{1-q^{2n}}\sin \left(\frac{n\pi z}{{\omega }_1}\right).$$

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Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\sin z$: sine function, $𝕃$: lattice, $q$: nome, $n$: integer, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Permalink:

http://dlmf.nist.gov/23.8.E2

Encodings:

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See also:

Annotations for §23.8(i), §23.8 and Ch.23

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§23.8(ii) Series of Cosecants and Cotangents

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23.8.4 $$\zeta \left(z\right)=\frac{{\eta }_{1}z}{{\omega }_1}+\frac{\pi }{2{\omega }_1}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\cot \left(\frac{\pi (z+2n{\omega }_3)}{2{\omega }_1}\right),$$

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Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $𝕃$: lattice, $n$: integer, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Referenced by:

§23.8(ii)

Permalink:

http://dlmf.nist.gov/23.8.E4

Encodings:

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See also:

Annotations for §23.8(ii), §23.8 and Ch.23

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2: 4.1 Special Notation

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$k,m,n$	integers.
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… ►The main functions treated in this chapter are the logarithm $\ln z$, $\mathrm{Ln}z$; the exponential $\exp z$, ${\mathrm{e}}^z$; the circular trigonometric (or just trigonometric) functions $\sin z$, $\cos z$, $\tan z$, $\csc z$, $\sec z$, $\cot z$; the inverse trigonometric functions $\arcsin z$, $\mathrm{Arcsin}z$, etc. ; 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. …

3: 4.29 Graphics

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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 $\mathrm{arccoth}x$. ($\mathrm{arctanh}x$ is complex when or $x>1$, and $\mathrm{arccoth}x$ is complex when .) Magnify

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4: 4.16 Elementary Properties

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Table 4.16.1: Signs of the trigonometric functions in the four quadrants.

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Quadrant	$\sin \theta ,\csc \theta$	$\cos \theta ,\sec \theta$	$\tan \theta ,\cot \theta$
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Table 4.16.2: Trigonometric functions: quarter periods and change of sign.

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$x$	$-\theta$	$\frac{1}{2}\pi \pm \theta$	$\pi \pm \theta$	$\frac{3}{2}\pi \pm \theta$	$2\pi \pm \theta$
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$\tan x$	$-\tan \theta$	$\mp \cot \theta$	$\pm \tan \theta$	$\mp \cot \theta$	$\pm \tan \theta$
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$\cot x$	$-\cot \theta$	$\mp \tan \theta$	$\pm \cot \theta$	$\mp \tan \theta$	$\pm \cot \theta$

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Table 4.16.3: Trigonometric functions: interrelations. …

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	$\sin \theta =a$	$\cos \theta =a$	$\tan \theta =a$	$\csc \theta =a$	$\sec \theta =a$	$\cot \theta =a$
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$\cot \theta$	$a^{-1}{(1-a^2)}^{1/2}$	$a{(1-a^2)}^{-1/2}$	$a^{-1}$	${(a^2-1)}^{1/2}$	${(a^2-1)}^{-1/2}$	$a$

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5: 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

Encodings:

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See also:

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

Encodings:

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See also:

Annotations for §4.28, §4.28 and Ch.4

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6: 4.24 Inverse Trigonometric Functions: Further Properties

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4.24.12 $$\frac{d}{dz}\mathrm{arccot}z=-\frac{1}{1+z^2}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arccot}z$: arccotangent function and $z$: complex variable

A&S Ref:

4.4.55

Permalink:

http://dlmf.nist.gov/4.24.E12

Encodings:

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See also:

Annotations for §4.24(ii), §4.24 and Ch.4

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4.24.17 $$\mathrm{Arctan}u\pm \mathrm{Arccot}v=\mathrm{Arctan}\left(\frac{uv\pm 1}{v\mp u}\right)=\mathrm{Arccot}\left(\frac{v\mp u}{uv\pm 1}\right).$$

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Symbols:

$\mathrm{Arccot}z$: general arccotangent function and $\mathrm{Arctan}z$: general arctangent function

A&S Ref:

4.4.36

Permalink:

http://dlmf.nist.gov/4.24.E17

Encodings:

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See also:

Annotations for §4.24(iii), §4.24 and Ch.4

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7: 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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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

Encodings:

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See also:

Annotations for §4.34 and Ch.4

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8: 4.19 Maclaurin Series and Laurent Series

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4.19.6 $$\cot z=\frac{1}{z}-\frac{z}{3}-\frac{z^3}{45}-\frac{2}{945}{z}^5-\mathrm{\cdots }-\frac{{(-1)}^{n-1}{2}^{2n}{B}_{2n}}{(2n)!}{z}^{2n-1}-\mathrm{\cdots },$$ ,

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $!$: factorial (as in $n!$), $n$: integer and $z$: complex variable

A&S Ref:

4.3.70

Permalink:

http://dlmf.nist.gov/4.19.E6

Encodings:

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See also:

Annotations for §4.19 and Ch.4

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9: 4.15 Graphics

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Figure 4.15.3: $\tan x$ and $\cot x$. Magnify

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Figure 4.15.4: $\arctan x$ and $\mathrm{arccot}x$. …$\mathrm{arccot}x$ is discontinuous at $x=0$. Magnify

… ►The corresponding surfaces for $\cos \left(x+\mathrm{i}y\right)$, $\cot \left(x+\mathrm{i}y\right)$, and $\sec \left(x+\mathrm{i}y\right)$ are similar. … ►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).

10: 4.14 Definitions and Periodicity

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4.14.7 $$\cot z=\frac{\cos z}{\sin z}=\frac{1}{\tan z}.$$

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Defines:

$\cot z$: cotangent function

Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.6

Referenced by:

§4.14, §4.45(i), §4.45(ii)

Permalink:

http://dlmf.nist.gov/4.14.E7

Encodings:

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Annotations for §4.14 and Ch.4

… ►The functions $\tan z$, $\csc z$, $\sec z$, and $\cot z$ are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7). …