hyperbolic secant function

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1: 4.1 Special Notation

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$k,m,n$	integers.
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… ►; 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: 22.10 Maclaurin Series

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22.10.7 $$\mathrm{sn}(z,k)=\tanh z-\frac{k_{}^{\prime }{}_{}{}^2}{4}(z-\sinh z\cosh z){\mathrm{sech}}^{2}z+O\left(k_{}^{\prime }{}_{}{}^4\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.1

Referenced by:

§22.10(ii)

Permalink:

http://dlmf.nist.gov/22.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.8 $$\mathrm{cn}(z,k)=\mathrm{sech}z+\frac{k_{}^{\prime }{}_{}{}^2}{4}(z-\sinh z\cosh z)\tanh z\mathrm{sech}z+O\left(k_{}^{\prime }{}_{}{}^4\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.2

Permalink:

http://dlmf.nist.gov/22.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.9 $$\mathrm{dn}(z,k)=\mathrm{sech}z+\frac{k_{}^{\prime }{}_{}{}^2}{4}(z+\sinh z\cosh z)\tanh z\mathrm{sech}z+O\left(k_{}^{\prime }{}_{}{}^4\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.15.3

Referenced by:

§22.10(ii)

Permalink:

http://dlmf.nist.gov/22.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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3: 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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§4.29(ii) Complex Arguments

►The conformal mapping $w=\sinh z$ is obtainable from Figure 4.15.7 by rotating both the $w$-plane and the $z$-plane through an angle $\frac{1}{2}\pi$, compare (4.28.8). ►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. …

4: 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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

Annotations for §4.31 and Ch.4

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5: 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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$\mathrm{sech}\theta$	${(1+a^2)}^{-1/2}$	$a^{-1}$	${(1-a^2)}^{1/2}$	$a{(1+a^2)}^{-1/2}$	$a$	$a^{-1}{(a^2-1)}^{1/2}$
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6: 4.28 Definitions and Periodicity

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4.28.6 $$\mathrm{sech}z=\frac{1}{\cosh z},$$

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Defines:

$\mathrm{sech}z$: hyperbolic secant function

Symbols:

$\cosh z$: hyperbolic cosine function and $z$: complex variable

A&S Ref:

4.5.5

Permalink:

http://dlmf.nist.gov/4.28.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.28 and Ch.4

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4.28.12 $$\sec \left(\mathrm{i}z\right)=\mathrm{sech}z,$$

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Symbols:

$\mathrm{sech}z$: hyperbolic secant function, $\mathrm{i}$: imaginary unit, $\sec z$: secant function and $z$: complex variable

A&S Ref:

4.5.11

Permalink:

http://dlmf.nist.gov/4.28.E12

Encodings:

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See also:

Annotations for §4.28, §4.28 and Ch.4

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7: 4.34 Derivatives and Differential Equations

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4.34.3 $$\frac{d}{dz}\tanh z={\mathrm{sech}}^{2}z,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{sech}z$: hyperbolic secant function, $\tanh z$: hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.5.73

Permalink:

http://dlmf.nist.gov/4.34.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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4.34.5 $$\frac{d}{dz}\mathrm{sech}z=-\mathrm{sech}z\tanh z,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{sech}z$: hyperbolic secant function, $\tanh z$: hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.5.75

Permalink:

http://dlmf.nist.gov/4.34.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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8: 4.37 Inverse Hyperbolic Functions

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4.37.5 $$\mathrm{Arcsech}z=\mathrm{Arccosh}\left(1/z\right),$$

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Defines:

$\mathrm{Arcsech}z$: general inverse hyperbolic secant function

Symbols:

$\mathrm{Arccosh}z$: general inverse hyperbolic cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E5

Encodings:

pMML, png, TeX

See also:

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

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4.37.8 $$\mathrm{arcsech}z=\mathrm{arccosh}\left(1/z\right).$$

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Defines:

$\mathrm{arcsech}z$: inverse hyperbolic secant function

Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function and $z$: complex variable

A&S Ref:

4.6.5

Permalink:

http://dlmf.nist.gov/4.37.E8

Encodings:

pMML, png, TeX

See also:

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

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4.37.14 $$\mathrm{arcsech}\left(-z\right)=\mp \pi \mathrm{i}+\mathrm{arcsech}z,$$ $\mathrm{\Im }z\gtrless 0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arcsech}z$: inverse hyperbolic secant function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E14

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). …

9: 28.20 Definitions and Basic Properties

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28.20.9 $${\mathrm{M}}_{\nu }^{(3)}(z,h)=H_{\nu }^{(1)}\left(2h\cosh z\right)\left(1+O\left(\mathrm{sech}z\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iii), §28.20 and Ch.28

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28.20.10 $${\mathrm{M}}_{\nu }^{(4)}(z,h)=H_{\nu }^{(2)}\left(2h\cosh z\right)\left(1+O\left(\mathrm{sech}z\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.20.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iii), §28.20 and Ch.28

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28.20.11 $${\mathrm{M}}_{\nu }^{(1)}(z,h)=J_{\nu }\left(2h\cosh z\right)+{\mathrm{e}}^{|\mathrm{\Im }\left(2h\cosh z\right)|}O\left({\left(\mathrm{sech}z\right)}^{3/2}\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\mathrm{\Im }$: imaginary part, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§28.33(ii)

Permalink:

http://dlmf.nist.gov/28.20.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iii), §28.20 and Ch.28

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28.20.12 $${\mathrm{M}}_{\nu }^{(2)}(z,h)=Y_{\nu }\left(2h\cosh z\right)+{\mathrm{e}}^{|\mathrm{\Im }\left(2h\cosh z\right)|}O\left({(\mathrm{sech}z)}^{3/2}\right),$$

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\mathrm{\Im }$: imaginary part, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§28.33(ii)

Permalink:

http://dlmf.nist.gov/28.20.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iii), §28.20 and Ch.28

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10: 10.19 Asymptotic Expansions for Large Order

§10.19 Asymptotic Expansions for Large Order

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§10.19(i) Asymptotic Forms

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§10.19(ii) Debye’s Expansions

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§10.19(iii) Transition Region

… ►See also §10.20(i).