cosecant function

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11: 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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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12: 4.15 Graphics

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4.15.3 $$\sec \left(x+\mathrm{i}y\right)=\csc \left(x+\frac{1}{2}\pi +\mathrm{i}y\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{i}$: imaginary unit, $\sec z$: secant function, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.15.E3

Encodings:

pMML, png, TeX

See also:

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

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13: 10.4 Connection Formulas

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10.4.5 $$J_{\nu }\left(z\right)=\csc \left(\nu \pi \right)\left(Y_{-\nu }\left(z\right)-Y_{\nu }\left(z\right)\cos \left(\nu \pi \right)\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.4

Permalink:

http://dlmf.nist.gov/10.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.4 and Ch.10

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10.4.7 $$H_{\nu }^{(1)}\left(z\right)=\mathrm{i}\csc \left(\nu \pi \right)\left({\mathrm{e}}^{-\nu \pi \mathrm{i}}{J}_{\nu }\left(z\right)-J_{-\nu }\left(z\right)\right)=\csc \left(\nu \pi \right)\left(Y_{-\nu }\left(z\right)-{\mathrm{e}}^{-\nu \pi \mathrm{i}}{Y}_{\nu }\left(z\right)\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.4, §10.8

Permalink:

http://dlmf.nist.gov/10.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.4 and Ch.10

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10.4.8 $$H_{\nu }^{(2)}\left(z\right)=\mathrm{i}\csc \left(\nu \pi \right)\left(J_{-\nu }\left(z\right)-{\mathrm{e}}^{\nu \pi \mathrm{i}}{J}_{\nu }\left(z\right)\right)=\csc \left(\nu \pi \right)\left(Y_{-\nu }\left(z\right)-{\mathrm{e}}^{\nu \pi \mathrm{i}}{Y}_{\nu }\left(z\right)\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.4, §10.8

Permalink:

http://dlmf.nist.gov/10.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.4 and Ch.10

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14: 10.34 Analytic Continuation

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10.34.2 $$K_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{-m\nu \pi \mathrm{i}}{K}_{\nu }\left(z\right)-\pi \mathrm{i}\sin \left(m\nu \pi \right)\csc \left(\nu \pi \right)I_{\nu }\left(z\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.31

Referenced by:

§10.27, §10.34, §10.34

Permalink:

http://dlmf.nist.gov/10.34.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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10.34.4 $$K_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=\csc \left(\nu \pi \right)\left(\pm \sin \left(m\nu \pi \right)K_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp \sin \left((m\mp 1)\nu \pi \right)K_{\nu }\left(z\right)\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.34, §10.40(i)

Permalink:

http://dlmf.nist.gov/10.34.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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15: 4.37 Inverse Hyperbolic Functions

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4.37.4 $$\mathrm{Arccsch}z=\mathrm{Arcsinh}\left(1/z\right),$$

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Defines:

$\mathrm{Arccsch}z$: general inverse hyperbolic cosecant function

Symbols:

$\mathrm{Arcsinh}z$: general inverse hyperbolic sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E4

Encodings:

pMML, png, TeX

See also:

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

… ► $\mathrm{Arcsinh}z$ and $\mathrm{Arccsch}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$. … ►

4.37.7 $$\mathrm{arccsch}z=\mathrm{arcsinh}\left(1/z\right),$$

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Defines:

$\mathrm{arccsch}z$: inverse hyperbolic cosecant function

Symbols:

$\mathrm{arcsinh}z$: inverse hyperbolic sine function and $z$: complex variable

A&S Ref:

4.6.4

Referenced by:

§4.37(iv), §4.45(i), §4.45(ii)

Permalink:

http://dlmf.nist.gov/4.37.E7

Encodings:

pMML, png, TeX

See also:

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

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4.37.13 $$\mathrm{arccsch}\left(-z\right)=-\mathrm{arccsch}z.$$

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Symbols:

$\mathrm{arccsch}z$: inverse hyperbolic cosecant function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E13

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). …

16: 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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4.17.1 $$\underset{z\to 0}{lim}\frac{\sin z}{z}=1,$$

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Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.74

Permalink:

http://dlmf.nist.gov/4.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

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4.17.2 $$\underset{z\to 0}{lim}\frac{\tan z}{z}=1.$$

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Symbols:

$\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.75

Permalink:

http://dlmf.nist.gov/4.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

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17: 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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$\mathrm{csch}\theta$	$a^{-1}$	${(a^2-1)}^{-1/2}$	$a^{-1}{(1-a^2)}^{1/2}$	$a$	$a{(1-a^2)}^{-1/2}$	${(a^2-1)}^{1/2}$
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18: 23.8 Trigonometric Series and Products

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23.8.1 $$\mathrm{\wp }\left(z\right)+\frac{{\eta }_1}{{\omega }_1}-\frac{{\pi }^2}{4{\omega }_1^2}{\csc }^2\left(\frac{\pi z}{2{\omega }_1}\right)=-\frac{2{\pi }^2}{{\omega }_1^2}\sum _{n=1}^{\mathrm{\infty }}\frac{n{q}^{2n}}{1-q^{2n}}\cos \left(\frac{n\pi z}{{\omega }_1}\right),$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine 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.E1

Encodings:

pMML, png, TeX

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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23.8.3 $$\mathrm{\wp }\left(z\right)=-\frac{{\eta }_1}{{\omega }_1}+\frac{{\pi }^2}{4{\omega }_1^2}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\csc }^2\left(\frac{\pi (z+2n{\omega }_3)}{2{\omega }_1}\right),$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $𝕃$: lattice, $n$: 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.E3

Encodings:

pMML, png, TeX

See also:

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

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23.8.5 $${\eta }_1=\frac{{\pi }^2}{2{\omega }_1}\left(\frac{1}{6}+\sum _{n=1}^{\mathrm{\infty }}{\csc }^2\left(\frac{n\pi {\omega }_3}{{\omega }_1}\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $n$: integer, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Permalink:

http://dlmf.nist.gov/23.8.E5

Encodings:

pMML, png, TeX

See also:

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

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19: 4.42 Solution of Triangles

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4.42.1 $$\sin A=\frac{a}{c}=\frac{1}{\csc A},$$

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Symbols:

$\csc z$: cosecant function, $\sin z$: sine function, $A$: angle, $a$: height and $c$: hypotenuse

A&S Ref:

4.3.147

Permalink:

http://dlmf.nist.gov/4.42.E1

Encodings:

pMML, png, TeX

See also:

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

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20: 19.11 Addition Theorems

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$$\gamma =(({\csc }^2\theta )-{\alpha }^2)(({\csc }^2\varphi )-{\alpha }^2)(({\csc }^2\psi )-{\alpha }^2),$$

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19.11.6_5 $$R_C(\gamma -\delta ,\gamma )=\frac{-1}{\sqrt{\delta }}\arctan \left(\frac{\sqrt{\delta }\sin \theta \sin \varphi \sin \psi }{{\alpha }^2-1-{\alpha }^2\cos \theta \cos \varphi \cos \psi }\right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $\arctan z$: arctangent function, $\sin z$: sine function, $\varphi$: real or complex argument, ${\alpha }^2$: real or complex parameter, $\theta$: amplitude, $\psi$: angle, $\gamma$ and $\delta$

Referenced by:

§19.11(i), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)

Permalink:

http://dlmf.nist.gov/19.11.E6_5

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.28):

This equation was given a number.

See also:

Annotations for §19.11(i), §19.11 and Ch.19

►Hence, care has to be taken with the multivalued functions in (19.11.5). … ►

$$\gamma =(1-{\alpha }^2)(({\csc }^2\theta )-{\alpha }^2)(({\csc }^2\varphi )-{\alpha }^2),$$

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$$\gamma ={(({\csc }^2\theta )-{\alpha }^2)}^2(({\csc }^2\psi )-{\alpha }^2),$$

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