hyperbolic cosecant function

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11: 24.7 Integral Representations

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24.7.2 $$B_{2n}={(-1)}^{n+1}4n{\int }_0^{\mathrm{\infty }}\frac{t^{2n-1}}{{\mathrm{e}}^{2\pi t}-1}dt={(-1)}^{n+1}2n{\int }_0^{\mathrm{\infty }}{t}^{2n-1}{\mathrm{e}}^{-\pi t}\mathrm{csch}\left(\pi t\right)dt,$$

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(i), §24.7 and Ch.24

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24.7.4 $$B_{2n}={(-1)}^{n+1}\pi {\int }_0^{\mathrm{\infty }}{t}^{2n}{\mathrm{csch}}^2\left(\pi t\right)dt,$$

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{csch}z$: hyperbolic cosecant function, $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(i), §24.7 and Ch.24

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12: 22.5 Special Values

§22.5 Special Values

… ►For the other nine functions ratios can be taken; compare (22.2.10). … ►

§22.5(ii) Limiting Values of $k$

… ►In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. … ►

13: 20.10 Integrals

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20.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_3\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_4\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=\frac{\mathrm{\ell }}{\sqrt{s}}\cosh \left(\beta \sqrt{s}\right)\mathrm{csch}\left(\mathrm{\ell }\sqrt{s}\right).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E5

Encodings:

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See also:

Annotations for §20.10(ii), §20.10 and Ch.20

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14: 4.23 Inverse Trigonometric Functions

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4.23.40 $$\mathrm{gd}\left(x\right)=2\arctan \left({\mathrm{e}}^x\right)-\frac{1}{2}\pi =\arcsin \left(\tanh x\right)=\mathrm{arccsc}\left(\coth x\right)=\arccos \left(\mathrm{sech}x\right)=\mathrm{arcsec}\left(\cosh x\right)=\arctan \left(\sinh x\right)=\mathrm{arccot}\left(\mathrm{csch}x\right).$$

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Symbols:

$\mathrm{gd}x$: Gudermannian function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\coth z$: hyperbolic cotangent function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $\mathrm{arccsc}z$: arccosecant function, $\arccos z$: arccosine function, $\mathrm{arccot}z$: arccotangent function, $\mathrm{arcsec}z$: arcsecant function, $\arcsin z$: arcsine function, $\arctan z$: arctangent function and $x$: real variable

Referenced by:

§4.23(viii), §4.40(ii)

Permalink:

http://dlmf.nist.gov/4.23.E40

Encodings:

pMML, png, TeX

See also:

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

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4.23.42 $${\mathrm{gd}}^{-1}\left(x\right)=\ln \tan \left(\frac{1}{2}x+\frac{1}{4}\pi \right)=\ln \left(\sec x+\tan x\right)=\mathrm{arcsinh}\left(\tan x\right)=\mathrm{arccsch}\left(\cot x\right)=\mathrm{arccosh}\left(\sec x\right)=\mathrm{arcsech}\left(\cos x\right)=\mathrm{arctanh}\left(\sin x\right)=\mathrm{arccoth}\left(\csc x\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $\cot z$: cotangent function, $\mathrm{arccsch}z$: inverse hyperbolic cosecant function, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{arccoth}z$: inverse hyperbolic cotangent function, $\mathrm{arcsech}z$: inverse hyperbolic secant function, $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\ln z$: principal branch of logarithm function, $\sec z$: secant function, $\sin z$: sine function, $\tan z$: tangent function and $x$: real variable

Referenced by:

§19.9(ii), §4.23(viii), §4.26(ii)

Permalink:

http://dlmf.nist.gov/4.23.E42

Encodings:

pMML, png, TeX

See also:

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

15: 4.35 Identities

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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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16: 10.7 Limiting Forms

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10.7.6 $$Y_{\mathrm{i}\nu }\left(z\right)=\frac{\mathrm{i}\mathrm{csch}\left(\nu \pi \right)}{\mathrm{\Gamma }\left(1-\mathrm{i}\nu \right)}{(\frac{1}{2}z)}^{-\mathrm{i}\nu }-\frac{\mathrm{i}\coth \left(\nu \pi \right)}{\mathrm{\Gamma }\left(1+\mathrm{i}\nu \right)}{(\frac{1}{2}z)}^{\mathrm{i}\nu }+{\mathrm{e}}^{|\nu \mathrm{ph}z|}o\left(1\right),$$ $\nu \in ℝ$ and $\nu \ne 0$.

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\coth z$: hyperbolic cotangent function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than, $\mathrm{ph}$: phase, $ℝ$: real line, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.7(i), §10.7(i)

Permalink:

http://dlmf.nist.gov/10.7.E6

Encodings:

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See also:

Annotations for §10.7(i), §10.7 and Ch.10

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17: 4.38 Inverse Hyperbolic Functions: Further Properties

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4.38.12 $$\frac{d}{dz}\mathrm{arccsch}z=\mp \frac{1}{z{(1+z^2)}^{1/2}},$$ $\mathrm{\Re }z\gtrless 0$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arccsch}z$: inverse hyperbolic cosecant function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.6.40

Permalink:

http://dlmf.nist.gov/4.38.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(ii), §4.38 and Ch.4

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18: 10.32 Integral Representations

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10.32.7 $$K_{\nu }\left(x\right)=\sec \left(\frac{1}{2}\nu \pi \right){\int }_0^{\mathrm{\infty }}\cos \left(x\sinh t\right)\cosh \left(\nu t\right)dt=\csc \left(\frac{1}{2}\nu \pi \right){\int }_0^{\mathrm{\infty }}\sin \left(x\sinh t\right)\sinh \left(\nu t\right)dt,$$ , $x>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $\sec z$: secant function, $\sin z$: sine function, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.6.22

Referenced by:

§10.32(i)

Permalink:

http://dlmf.nist.gov/10.32.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(i), §10.32 and Ch.10

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19: 4.45 Methods of Computation

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Hyperbolic and Inverse Hyperbolic Functions

►The hyperbolic functions can be computed directly from the definitions (4.28.1)–(4.28.7). The inverses $\mathrm{arcsinh}$, $\mathrm{arccosh}$, and $\mathrm{arctanh}$ can be computed from the logarithmic forms given in §4.37(iv), with real arguments. For $\mathrm{arccsch}$, $\mathrm{arcsech}$, and $\mathrm{arccoth}$ we have (4.37.7)–(4.37.9). … ►Similarly for the hyperbolic and inverse hyperbolic functions; compare (4.28.1)–(4.28.7), §4.37(iv), and (4.37.7)–(4.37.9). …

20: 19.7 Connection Formulas

… ►With $\sinh \varphi =\tan \psi$, … ►The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ when ${\alpha }^2>{\csc }^2\varphi$ (see (19.6.5) for the complete case). Let $c={\csc }^2\varphi \ne {\alpha }^2$. … ►The second relation maps each hyperbolic region onto itself and each circular region onto the other: … ►The third relation (missing from the literature of Legendre’s integrals) maps each circular region onto the other and each hyperbolic region onto the other: …