cosecant function

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1: 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. …

2: 4.28 Definitions and Periodicity

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4.28.5 $$\mathrm{csch}z=\frac{1}{\sinh z},$$

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Defines:

$\mathrm{csch}z$: hyperbolic cosecant function

Symbols:

$\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.4

Permalink:

http://dlmf.nist.gov/4.28.E5

Encodings:

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See also:

Annotations for §4.28 and Ch.4

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4.28.11 $$\csc \left(\mathrm{i}z\right)=-\mathrm{i}\mathrm{csch}z,$$

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Symbols:

$\csc z$: cosecant function, $\mathrm{csch}z$: hyperbolic cosecant function, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.5.10

Permalink:

http://dlmf.nist.gov/4.28.E11

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Annotations for §4.28, §4.28 and Ch.4

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3: 4.22 Infinite Products and Partial Fractions

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4.22.4 $${\csc }^{2}z=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{{(z-n\pi )}^2},$$

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Symbols:

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

A&S Ref:

4.3.92

Permalink:

http://dlmf.nist.gov/4.22.E4

Encodings:

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See also:

Annotations for §4.22 and Ch.4

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4.22.5 $$\csc z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{z^2-n^2{\pi }^2}.$$

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Symbols:

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

A&S Ref:

4.3.93

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/4.22.E5

Encodings:

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Annotations for §4.22 and Ch.4

4: 4.36 Infinite Products and Partial Fractions

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4.36.4 $${\mathrm{csch}}^{2}z=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{{(z-n\pi \mathrm{i})}^2},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{csch}z$: hyperbolic cosecant function, $\mathrm{i}$: imaginary unit, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.36.E4

Encodings:

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See also:

Annotations for §4.36 and Ch.4

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4.36.5 $$\mathrm{csch}z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{z^2+n^2{\pi }^2}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{csch}z$: hyperbolic cosecant function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.36.E5

Encodings:

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See also:

Annotations for §4.36 and Ch.4

5: 4.14 Definitions and Periodicity

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4.14.5 $$\csc z=\frac{1}{\sin z},$$

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Defines:

$\csc z$: cosecant function

Symbols:

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

A&S Ref:

4.3.4

Permalink:

http://dlmf.nist.gov/4.14.E5

Encodings:

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See also:

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). …

6: 4.29 Graphics

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§4.29(i) Real Arguments

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Figure 4.29.5: $\mathrm{csch}x$ and $\mathrm{sech}x$. Magnify

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Figure 4.29.6: Principal values of $\mathrm{arccsch}x$ and $\mathrm{arcsech}x$. … 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. …

7: 4.16 Elementary Properties

§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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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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8: 4.18 Inequalities

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4.18.7 $$|\csc z|\le \mathrm{csch}|y|,$$

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Symbols:

$\csc z$: cosecant function, $\mathrm{csch}z$: hyperbolic cosecant function, $y$: real variable and $z$: complex variable

A&S Ref:

4.3.85

Permalink:

http://dlmf.nist.gov/4.18.E7

Encodings:

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Annotations for §4.18, §4.18 and Ch.4

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9: 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:

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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:

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See also:

Annotations for §4.20 and Ch.4

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10: 4.31 Special Values and Limits

§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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4.31.1 $$\underset{z\to 0}{lim}\frac{\sinh z}{z}=1,$$

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Symbols:

$\sinh z$: hyperbolic sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E1

Encodings:

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See also:

Annotations for §4.31 and Ch.4

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4.31.2 $$\underset{z\to 0}{lim}\frac{\tanh z}{z}=1,$$

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Symbols:

$\tanh z$: hyperbolic tangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E2

Encodings:

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See also:

Annotations for §4.31 and Ch.4

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