hyperbolic secant function

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11: 24.7 Integral Representations

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24.7.1 $$B_{2n}={(-1)}^{n+1}\frac{4n}{1-2^{1-2n}}{\int }_0^{\mathrm{\infty }}\frac{t^{2n-1}}{{\mathrm{e}}^{2\pi t}+1}dt={(-1)}^{n+1}\frac{2n}{1-2^{1-2n}}{\int }_0^{\mathrm{\infty }}{t}^{2n-1}{\mathrm{e}}^{-\pi t}\mathrm{sech}\left(\pi t\right)dt,$$

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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, $\mathrm{sech}z$: hyperbolic secant function, $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E1

Encodings:

pMML, png, TeX

See also:

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

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24.7.3 $$B_{2n}={(-1)}^{n+1}\frac{\pi }{1-2^{1-2n}}{\int }_0^{\mathrm{\infty }}{t}^{2n}{\mathrm{sech}}^2\left(\pi t\right)dt,$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/24.7.E3

Encodings:

pMML, png, TeX

See also:

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

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24.7.6 $$E_{2n}={(-1)}^n{2}^{2n+1}{\int }_0^{\mathrm{\infty }}{t}^{2n}\mathrm{sech}\left(\pi t\right)dt.$$

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Symbols:

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

Referenced by:

§24.7(i)

Permalink:

http://dlmf.nist.gov/24.7.E6

Encodings:

pMML, png, TeX

See also:

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

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12: 4.23 Inverse Trigonometric Functions

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4.23.39 $$\mathrm{gd}\left(x\right)={\int }_0^x\mathrm{sech}tdt,$$ .

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Defines:

$\mathrm{gd}x$: Gudermannian function

Symbols:

$dx$: differential of $x$ , $\mathrm{sech}z$: hyperbolic secant function, $\int$: integral and $x$: real variable

Referenced by:

§4.40(ii)

Permalink:

http://dlmf.nist.gov/4.23.E39

Encodings:

pMML, png, TeX

See also:

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

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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

13: 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. … ►

14: 4.40 Integrals

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4.40.5 $$\int \mathrm{sech}xdx=\mathrm{gd}\left(x\right).$$

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Symbols:

$\mathrm{gd}x$: Gudermannian function, $dx$: differential of $x$ , $\mathrm{sech}z$: hyperbolic secant function, $\int$: integral and $x$: real variable

A&S Ref:

4.5.81

Permalink:

http://dlmf.nist.gov/4.40.E5

Encodings:

pMML, png, TeX

See also:

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

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4.40.15 $$\int \mathrm{arcsech}xdx=x\mathrm{arcsech}x+\arcsin x,$$ ,

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Symbols:

$dx$: differential of $x$ , $\mathrm{arcsech}z$: inverse hyperbolic secant function, $\int$: integral, $\arcsin z$: arcsine function and $x$: real variable

A&S Ref:

4.6.47 (modified. The first printing had an error.)

Permalink:

http://dlmf.nist.gov/4.40.E15

Encodings:

pMML, png, TeX

See also:

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

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15: 20.10 Integrals

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20.10.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_1\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_2\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=-\frac{\mathrm{\ell }}{\sqrt{s}}\sinh \left(\beta \sqrt{s}\right)\mathrm{sech}\left(\mathrm{\ell }\sqrt{s}\right),$$

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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, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E4

Encodings:

pMML, png, TeX

See also:

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

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16: 22.16 Related Functions

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22.16.8 $$\mathrm{am}(x,k)=\mathrm{gd}x-\frac{1}{4}k_{}^{\prime }{}_{}{}^2(x-\sinh x\cosh x)\mathrm{sech}x+O\left(k_{}^{\prime }{}_{}{}^4\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{gd}x$: Gudermannian function, $\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.4

Permalink:

http://dlmf.nist.gov/22.16.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

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17: 5.6 Inequalities

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5.6.7 $$|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|\ge {(\mathrm{sech}\left(\pi y\right))}^{1/2}\mathrm{\Gamma }\left(x\right),$$ $x\ge \frac{1}{2}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{sech}z$: hyperbolic secant function, $\mathrm{i}$: imaginary unit, $x$: real variable and $y$: real variable

Referenced by:

§5.6(ii)

Permalink:

http://dlmf.nist.gov/5.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(ii), §5.6 and Ch.5

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18: 14.5 Special Values

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14.5.26 $${𝑸}_{\frac{1}{2}}\left(\cosh \xi \right)=2{\pi }^{-1/2}\cosh \xi \mathrm{sech}\left(\frac{1}{2}\xi \right)K\left(\mathrm{sech}\left(\frac{1}{2}\xi \right)\right)-4{\pi }^{-1/2}\cosh \left(\frac{1}{2}\xi \right)E\left(\mathrm{sech}\left(\frac{1}{2}\xi \right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $\xi >0$: variable

A&S Ref:

8.13.7 (in slightly different form)

Permalink:

http://dlmf.nist.gov/14.5.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(v), §14.5 and Ch.14

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19: 4.35 Identities

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4.35.12 $${\mathrm{sech}}^{2}z=1-{\tanh }^{2}z,$$

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Symbols:

$\mathrm{sech}z$: hyperbolic secant function, $\tanh z$: hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.5.17

Permalink:

http://dlmf.nist.gov/4.35.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.35(ii), §4.35 and Ch.4

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20: 15.9 Relations to Other Functions

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15.9.14 $${\mathrm{\Phi }}_{\lambda }^{(\alpha ,\beta )}(t)={(2\cosh t)}^{\mathrm{i}\lambda -\alpha -\beta -1}F(\genfrac{}{}{0pt}{}{\frac{1}{2}(\alpha +\beta +1-\mathrm{i}\lambda ),\frac{1}{2}(\alpha -\beta +1-\mathrm{i}\lambda )}{1-\mathrm{i}\lambda };{\mathrm{sech}}^{2}t).$$

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Defines:

${\mathrm{\Phi }}_{\lambda }^{(\alpha ,\beta )}(t)$: function (locally)

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\mathrm{i}$: imaginary unit and $(a,b)$: open interval

Permalink:

http://dlmf.nist.gov/15.9.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(ii), §15.9 and Ch.15

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