series of cosecants or cotangents

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21: 4.13 Lambert $W$-Function

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4.13.15 $$W_0\left(z\right)=\frac{z}{\pi }{\int }_0^{\pi }\frac{{\left(1-t\cot t\right)}^2+t^2}{z+t{\mathrm{e}}^{-t\cot t}\csc t}dt.$$

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.13.E15

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

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4.13.16 $$W_0\left(z\right)=\frac{1}{\pi }{\int }_0^{\pi }\ln \left(1+z\frac{\sin t}{t}{\mathrm{e}}^{t\cot t}\right)dt.$$

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function, $\sin z$: sine function and $z$: complex variable

Notes:

See Mező (2020, formula (3)).

Permalink:

http://dlmf.nist.gov/4.13.E16

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

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22: 4.40 Integrals

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4.40.6 $$\int \coth xdx=\ln \left(\sinh x\right),$$ .

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Symbols:

$dx$: differential of $x$ , $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.82

Permalink:

http://dlmf.nist.gov/4.40.E6

Encodings:

pMML, png, TeX

See also:

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

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4.40.7 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-x}\frac{\sin \left(ax\right)}{\sinh x}dx=\frac{1}{2}\pi \coth \left(\frac{1}{2}\pi a\right)-\frac{1}{a},$$ $a\ne 0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $a$: real or complex constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.40.E7

Encodings:

pMML, png, TeX

See also:

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

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4.40.16 $$\int \mathrm{arccoth}xdx=x\mathrm{arccoth}x+\frac{1}{2}\ln \left(x^2-1\right),$$ .

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Symbols:

$dx$: differential of $x$ , $\mathrm{arccoth}z$: inverse hyperbolic cotangent function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.6.48 (modified)

Permalink:

http://dlmf.nist.gov/4.40.E16

Encodings:

pMML, png, TeX

See also:

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

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23: 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:

pMML, png, TeX

See also:

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). … ►For example, $\mathrm{arcsech}a=\mathrm{arccoth}\left({(1-a^2)}^{-1/2}\right)$.

24: 22.11 Fourier and Hyperbolic Series

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22.11.9 $$\mathrm{cs}(z,k)-\frac{\pi }{2K}\cot \zeta =-\frac{2\pi }{K}\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}\sin \left(2n\zeta \right)}{1+q^{2n}},$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cot z$: cotangent function, $q$: nome, $\sin z$: sine function, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.12

Permalink:

http://dlmf.nist.gov/22.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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25: 4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

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$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
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26: 4.23 Inverse Trigonometric Functions

… ► $\mathrm{Arctan}z$ and $\mathrm{Arccot}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$. … ►The principal values of the inverse cosecant, secant, and cotangent are given by … ►

4.23.15 $$\mathrm{arccot}\left(-z\right)=-\mathrm{arccot}z,$$ $z\ne \pm \mathrm{i}$.

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Symbols:

$\mathrm{i}$: imaginary unit, $\mathrm{arccot}z$: arccotangent function and $z$: complex variable

A&S Ref:

4.4.19 (Early printings had an error.)

Permalink:

http://dlmf.nist.gov/4.23.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(iii), §4.23 and Ch.4

… ►For the corresponding results for $\mathrm{arccsc}z$, $\mathrm{arcsec}z$, and $\mathrm{arccot}z$, use (4.23.7)–(4.23.9). … ►For example, from the heading and last entry in the penultimate column we have $\mathrm{arcsec}a=\mathrm{arccot}\left({(a^2-1)}^{-1/2}\right)$. …

27: 10.19 Asymptotic Expansions for Large Order

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$$J_{\nu }\left(\nu \mathrm{sech}\alpha \right)\sim \frac{{\mathrm{e}}^{\nu (\tanh \alpha -\alpha )}}{{(2\pi \nu \tanh \alpha )}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}\frac{U_k(\coth \alpha )}{{\nu }^k},$$

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$$J_{\nu }\left(\nu \sec \beta \right)\sim {\left(\frac{2}{\pi \nu \tan \beta }\right)}^{\frac{1}{2}}\left(\cos \xi \sum _{k=0}^{\mathrm{\infty }}\frac{U_{2k}(\mathrm{i}\cot \beta )}{{\nu }^{2k}}-\mathrm{i}\sin \xi \sum _{k=0}^{\mathrm{\infty }}\frac{U_{2k+1}(\mathrm{i}\cot \beta )}{{\nu }^{2k+1}}\right),$$

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$$Y_{\nu }\left(\nu \sec \beta \right)\sim {\left(\frac{2}{\pi \nu \tan \beta }\right)}^{\frac{1}{2}}\left(\sin \xi \sum _{k=0}^{\mathrm{\infty }}\frac{U_{2k}(\mathrm{i}\cot \beta )}{{\nu }^{2k}}+\mathrm{i}\cos \xi \sum _{k=0}^{\mathrm{\infty }}\frac{U_{2k+1}(\mathrm{i}\cot \beta )}{{\nu }^{2k+1}}\right),$$

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$$J_{\nu }^{\prime }\left(\nu \sec \beta \right)\sim {\left(\frac{\sin \left(2\beta \right)}{\pi \nu }\right)}^{\frac{1}{2}}\left(-\sin \xi \sum _{k=0}^{\mathrm{\infty }}\frac{V_{2k}(\mathrm{i}\cot \beta )}{{\nu }^{2k}}-\mathrm{i}\cos \xi \sum _{k=0}^{\mathrm{\infty }}\frac{V_{2k+1}(\mathrm{i}\cot \beta )}{{\nu }^{2k+1}}\right),$$

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$$Y_{\nu }^{\prime }\left(\nu \sec \beta \right)\sim {\left(\frac{\sin \left(2\beta \right)}{\pi \nu }\right)}^{\frac{1}{2}}\left(\cos \xi \sum _{k=0}^{\mathrm{\infty }}\frac{V_{2k}(\mathrm{i}\cot \beta )}{{\nu }^{2k}}-\mathrm{i}\sin \xi \sum _{k=0}^{\mathrm{\infty }}\frac{V_{2k+1}(\mathrm{i}\cot \beta )}{{\nu }^{2k+1}}\right).$$

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28: 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:

pMML, png, TeX

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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

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

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29: 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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30: 23.12 Asymptotic Approximations

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23.12.2 $$\zeta \left(z\right)=\frac{{\pi }^2}{4{\omega }_1^2}\left(\frac{z}{3}+\frac{2{\omega }_1}{\pi }\cot \left(\frac{\pi z}{2{\omega }_1}\right)-8\left(z-\frac{{\omega }_1}{\pi }\sin \left(\frac{\pi z}{{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): 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, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§23.12, Erratum (V1.0.23) for Equation (23.12.2)

Permalink:

http://dlmf.nist.gov/23.12.E2

Encodings:

pMML, png, TeX

Correction (effective with 1.0.23):

The factor of 2, previously omitted from the denominator of the argument of the $\cot$ function has been inserted.

Suggested 2019-05-27 by Blagoje Oblak

See also:

Annotations for §23.12 and Ch.23

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