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21: 4.30 Elementary Properties

§4.30 Elementary Properties

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Table 4.30.1: Hyperbolic functions: interrelations. …

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	$\sinh \theta =a$	$\cosh \theta =a$	$\tanh \theta =a$	$\mathrm{csch}\theta =a$	$\mathrm{sech}\theta =a$	$\coth \theta =a$
$\sinh \theta$	$a$	${(a^2-1)}^{1/2}$	$a{(1-a^2)}^{-1/2}$	$a^{-1}$	$a^{-1}{(1-a^2)}^{1/2}$	${(a^2-1)}^{-1/2}$
$\cosh \theta$	${(1+a^2)}^{1/2}$	$a$	${(1-a^2)}^{-1/2}$	$a^{-1}{(1+a^2)}^{1/2}$	$a^{-1}$	$a{(a^2-1)}^{-1/2}$
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22: 36.5 Stokes Sets

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§36.5(iii) Umbilics

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Hyperbolic Umbilic Stokes Set (Codimension three)

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Figure 36.5.5: Elliptic umbilic catastrophe with $z=\mathrm{constant}$. … Magnify

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Figure 36.5.9: Sheets of the Stokes surface for the hyperbolic umbilic catastrophe (colored and with mesh) and the bifurcation set (gray). Magnify

23: 11.8 Analogs to Kelvin Functions

§11.8 Analogs to Kelvin Functions

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24: 36.4 Bifurcation Sets

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$x=\frac{9}{20}{z}^2.$

… ►Hyperbolic umbilic bifurcation set (codimension three): … ►Hyperbolic umbilic cusp line (rib): … ►

§36.4(ii) Visualizations

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Figure 36.4.4: Bifurcation set of hyperbolic umbilic catastrophe. Magnify

25: 4.39 Continued Fractions

§4.39 Continued Fractions

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4.39.1 $$\tanh z=\frac{z}{1+}\frac{z^2}{3+}\frac{z^2}{5+}\frac{z^2}{7+}\mathrm{\cdots },$$ $z\ne \pm \frac{1}{2}\pi \mathrm{i},\pm \frac{3}{2}\pi \mathrm{i},\mathrm{\dots }$.

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Symbols:

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

A&S Ref:

4.5.70

Permalink:

http://dlmf.nist.gov/4.39.E1

Encodings:

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See also:

Annotations for §4.39 and Ch.4

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4.39.2 $$\frac{\mathrm{arcsinh}z}{\sqrt{1+z^2}}=\frac{z}{1+}\frac{1\cdot 2z^2}{3+}\frac{1\cdot 2z^2}{5+}\frac{3\cdot 4z^2}{7+}\frac{3\cdot 4z^2}{9+}\mathrm{\cdots },$$

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Symbols:

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

A&S Ref:

4.6.36

Permalink:

http://dlmf.nist.gov/4.39.E2

Encodings:

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See also:

Annotations for §4.39 and Ch.4

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4.39.3 $$\mathrm{arctanh}z=\frac{z}{1-}\frac{z^2}{3-}\frac{4z^2}{5-}\frac{9z^2}{7-}\mathrm{\cdots },$$

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Symbols:

$\mathrm{arctanh}z$: inverse hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.6.35

Permalink:

http://dlmf.nist.gov/4.39.E3

Encodings:

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See also:

Annotations for §4.39 and Ch.4

… ►For these and other continued fractions involving inverse hyperbolic functions see Lorentzen and Waadeland (1992, pp. 569–571). …

26: 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:

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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:

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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:

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See also:

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

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27: 4.38 Inverse Hyperbolic Functions: Further Properties

§4.38 Inverse Hyperbolic Functions: Further Properties

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§4.38(i) Power Series

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§4.38(ii) Derivatives

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§4.38(iii) Addition Formulas

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4.38.19 $$\mathrm{Arctanh}u\pm \mathrm{Arccoth}v=\mathrm{Arctanh}\left(\frac{uv\pm 1}{v\pm u}\right)=\mathrm{Arccoth}\left(\frac{v\pm u}{uv\pm 1}\right).$$

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Symbols:

$\mathrm{Arccoth}z$: general inverse hyperbolic cotangent function and $\mathrm{Arctanh}z$: general inverse hyperbolic tangent function

A&S Ref:

4.6.30

Permalink:

http://dlmf.nist.gov/4.38.E19

Encodings:

pMML, png, TeX

See also:

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

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28: 4.46 Tables

§4.46 Tables

… ►For 10S values of the first five complex roots of $\sin z=az$, $\cos z=az$, and $\cosh z=az$, for selected positive values of $a$, see Fettis (1976). …

29: 20 Theta Functions

Chapter 20 Theta Functions

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30: 7.8 Inequalities

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7.8.5 $x\ge 0$.

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Symbols:

$𝖬\left(x\right)$: Mills’ ratio and $x$: real variable

Referenced by:

§7.8

Permalink:

http://dlmf.nist.gov/7.8.E5

Encodings:

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See also:

Annotations for §7.8 and Ch.7

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7.8.7 $x>0$.

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Symbols:

$F\left(z\right)$: Dawson’s integral, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\int$: integral and $x$: real variable

Referenced by:

Erratum (V1.1.5) for Additions

Permalink:

http://dlmf.nist.gov/7.8.E7

Encodings:

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Addition (effective with 1.1.5):

A new inequality was added.

See also:

Annotations for §7.8 and Ch.7

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