hyperbolic cosine function

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11: 28.32 Mathematical Applications

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28.32.3 $$\frac{{\partial }^{2}V}{{\partial \xi }^2}+\frac{{\partial }^{2}V}{{\partial \eta }^2}+\frac{1}{2}{c}^2{k}^2(\cosh \left(2\xi \right)-\cos \left(2\eta \right))V=0.$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\eta$: variable, $\xi$: variable, $k$: parameter and $V(\xi ,\eta )$: solution

Referenced by:

§28.32(i), §28.33(ii)

Permalink:

http://dlmf.nist.gov/28.32.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

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12: 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

… ►Each is two-valued on the corresponding cut(s), and each is real on the part of the real axis that remains after deleting the intersections with the corresponding cuts. … ►

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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4.37.27 $$z=\cosh w,$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/4.37.E27

Encodings:

pMML, png, TeX

See also:

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

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4.37.30 $$w=\mathrm{Arccosh}z=\pm \mathrm{arccosh}z+2k\pi \mathrm{i},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\mathrm{Arccosh}z$: general inverse hyperbolic cosine function, $k$: integer and $z$: complex variable

A&S Ref:

4.6.9

Permalink:

http://dlmf.nist.gov/4.37.E30

Encodings:

pMML, png, TeX

See also:

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

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13: 4.34 Derivatives and Differential Equations

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4.34.1 $$\frac{d}{dz}\sinh z=\cosh z,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.71

Permalink:

http://dlmf.nist.gov/4.34.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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4.34.2 $$\frac{d}{dz}\cosh z=\sinh z,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.72

Permalink:

http://dlmf.nist.gov/4.34.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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4.34.11 $$w=A\cosh \left(az\right)+B\sinh \left(az\right),$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $a$: real or complex constant, $z$: complex variable, $w$: solution, $A$: arbitrary constant and $B$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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4.34.13 $$w=(1/a)\cosh \left(az+c\right),$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.34 and Ch.4

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14: 14.19 Toroidal (or Ring) Functions

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14.19.4 $$P_{n-\frac{1}{2}}^m\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(n+m+\frac{1}{2}\right){(\sinh \xi )}^m}{2^m{\pi }^{1/2}\mathrm{\Gamma }\left(n-m+\frac{1}{2}\right)\mathrm{\Gamma }\left(m+\frac{1}{2}\right)}{\int }_0^{\pi }\frac{{(\sin \varphi )}^{2m}}{{(\cosh \xi +\cos \varphi \sinh \xi )}^{n+m+(1/2)}}d\varphi ,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer and $\xi >0$: variable

A&S Ref:

8.11.2

Permalink:

http://dlmf.nist.gov/14.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.19(iii), §14.19 and Ch.14

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14.19.5 $${𝑸}_{n-\frac{1}{2}}^m\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(n+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(n+m+\frac{1}{2}\right)\mathrm{\Gamma }\left(n-m+\frac{1}{2}\right)}{\int }_0^{\mathrm{\infty }}\frac{\cosh \left(mt\right)}{{(\cosh \xi +\cosh t\sinh \xi )}^{n+(1/2)}}dt,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $m$: nonnegative integer, $n$: nonnegative integer and $\xi >0$: variable

A&S Ref:

8.11.4 (modified and corrected)

Permalink:

http://dlmf.nist.gov/14.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.19(iii), §14.19 and Ch.14

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14.19.6 $${𝑸}_{-\frac{1}{2}}^{\mu }\left(\cosh \xi \right)+2\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\mu +n+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)}{𝑸}_{n-\frac{1}{2}}^{\mu }\left(\cosh \xi \right)\cos \left(n\varphi \right)=\frac{{\left(\frac{1}{2}\pi \right)}^{1/2}{\left(\sinh \xi \right)}^{\mu }}{{\left(\cosh \xi -\cos \varphi \right)}^{\mu +(1/2)}},$$ $\mathrm{\Re }\mu >-\frac{1}{2}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $\mu$: general order and $\xi >0$: variable

Permalink:

http://dlmf.nist.gov/14.19.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.19(iv), §14.19 and Ch.14

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14.19.7 $$P_{n-\frac{1}{2}}^m\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(n+m+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(n-m+\frac{1}{2}\right)}{\left(\frac{2}{\pi \sinh \xi }\right)}^{1/2}{𝑸}_{m-\frac{1}{2}}^n\left(\coth \xi \right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $m$: nonnegative integer, $n$: nonnegative integer and $\xi >0$: variable

Permalink:

http://dlmf.nist.gov/14.19.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §14.19(v), §14.19 and Ch.14

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14.19.8 $${𝑸}_{n-\frac{1}{2}}^m\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(m-n+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(m+n+\frac{1}{2}\right)}{\left(\frac{\pi }{2\sinh \xi }\right)}^{1/2}{P}_{m-\frac{1}{2}}^n\left(\coth \xi \right).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $m$: nonnegative integer, $n$: nonnegative integer and $\xi >0$: variable

Permalink:

http://dlmf.nist.gov/14.19.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §14.19(v), §14.19 and Ch.14

15: 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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	$\sinh \theta =a$	$\cosh \theta =a$	$\tanh \theta =a$	$\mathrm{csch}\theta =a$	$\mathrm{sech}\theta =a$	$\coth \theta =a$
…
$\cosh \theta$	${(1+a^2)}^{1/2}$	$a$	${(1-a^2)}^{-1/2}$	$a^{-1}{(1+a^2)}^{1/2}$	$a^{-1}$	$a{(a^2-1)}^{-1/2}$
…

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16: 14.25 Integral Representations

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14.25.1 $$P_{\nu }^{-\mu }\left(z\right)=\frac{{\left(z^2-1\right)}^{\mu /2}}{2^{\nu }\mathrm{\Gamma }\left(\mu -\nu \right)\mathrm{\Gamma }\left(\nu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\nu +1}}{{(z+\cosh t)}^{\nu +\mu +1}}dt,$$ $\mathrm{\Re }\mu >\mathrm{\Re }\nu >-1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.25

Permalink:

http://dlmf.nist.gov/14.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.25 and Ch.14

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14.25.2 $${𝑸}_{\nu }^{\mu }\left(z\right)=\frac{{\pi }^{1/2}{\left(z^2-1\right)}^{\mu /2}}{2^{\mu }\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{\left(z+{(z^2-1)}^{1/2}\cosh t\right)}^{\nu +\mu +1}}dt,$$ $\mathrm{\Re }\left(\nu +1\right)>\mathrm{\Re }\mu >-\frac{1}{2}$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.25

Permalink:

http://dlmf.nist.gov/14.25.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.25 and Ch.14

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17: 4.29 Graphics

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§4.29(i) Real Arguments

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►►

Figure 4.29.1: $\sinh x$ and $\cosh x$. 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. …

18: 4.40 Integrals

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4.40.1 $$\int \sinh xdx=\cosh x,$$

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Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral and $x$: real variable

A&S Ref:

4.5.77

Permalink:

http://dlmf.nist.gov/4.40.E1

Encodings:

pMML, png, TeX

See also:

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

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4.40.2 $$\int \cosh xdx=\sinh x,$$

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Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral and $x$: real variable

A&S Ref:

4.5.78

Permalink:

http://dlmf.nist.gov/4.40.E2

Encodings:

pMML, png, TeX

See also:

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

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4.40.3 $$\int \tanh xdx=\ln \left(\cosh x\right).$$

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Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.79

Permalink:

http://dlmf.nist.gov/4.40.E3

Encodings:

pMML, png, TeX

See also:

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

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4.40.9 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{ax}}{{\left(\cosh \left(\frac{1}{2}x\right)\right)}^2}dx=\frac{4\pi a}{\sin \left(\pi a\right)},$$ ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $a$: real or complex constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.40.E9

Encodings:

pMML, png, TeX

See also:

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

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4.40.12 $$\int \mathrm{arccosh}xdx=x\mathrm{arccosh}x-{(x^2-1)}^{1/2},$$ ,

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Symbols:

$dx$: differential of $x$ , $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\int$: integral and $x$: real variable

A&S Ref:

4.6.44 (modified)

Permalink:

http://dlmf.nist.gov/4.40.E12

Encodings:

pMML, png, TeX

See also:

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

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19: 28.26 Asymptotic Approximations for Large $q$

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28.26.1 $${\mathrm{Mc}}_m^{(3)}(z,h)=\frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}\left({\mathrm{Fc}}_m(z,h)-\mathrm{i}{\mathrm{Gc}}_m(z,h)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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28.26.2 $$\mathrm{i}{\mathrm{Ms}}_{m+1}^{(3)}(z,h)=\frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}\left({\mathrm{Fs}}_m(z,h)-\mathrm{i}{\mathrm{Gs}}_m(z,h)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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28.26.4 $${\mathrm{Fc}}_m(z,h)\sim 1+\frac{s}{8h{\cosh }^{2}z}+\frac{1}{2^{11}{h}^2}\left(\frac{s^4+86s^2+105}{{\cosh }^{4}z}-\frac{s^4+22s^2+57}{{\cosh }^{2}z}\right)+\frac{1}{2^{14}{h}^3}\left(-\frac{s^5+14s^3+33s}{{\cosh }^{2}z}-\frac{2s^5+124s^3+1122s}{{\cosh }^{4}z}+\frac{3s^5+290s^3+1627s}{{\cosh }^{6}z}\right)+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\cosh z$: hyperbolic cosine function, $m$: integer, $h$: parameter and $z$: complex variable

A&S Ref:

20.9.9 (in slightly different notation)

Referenced by:

§28.26(i)

Permalink:

http://dlmf.nist.gov/28.26.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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28.26.5 $${\mathrm{Gc}}_m(z,h)\sim \frac{\sinh z}{{\cosh }^{2}z}\left(\frac{s^2+3}{2^{5}h}+\frac{1}{2^9{h}^2}\left(s^3+3s+\frac{4s^3+44s}{{\cosh }^{2}z}\right)+\frac{1}{2^{14}{h}^3}\left(5s^4+34s^2+9-\frac{s^6-47s^4+667s^2+2835}{12{\cosh }^{2}z}+\frac{s^6+505s^4+12139s^2+10395}{12{\cosh }^{4}z}\right)\right)+\mathrm{\cdots }.$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $m$: integer, $h$: parameter and $z$: complex variable

A&S Ref:

20.9.10 (in slightly different notation)

Referenced by:

§28.26(i)

Permalink:

http://dlmf.nist.gov/28.26.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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20: 10.64 Integral Representations

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10.64.1 $${\mathrm{ber}}_n\left(x\sqrt{2}\right)=\frac{{(-1)}^n}{\pi }{\int }_0^{\pi }\cos \left(x\sin t-nt\right)\cosh \left(x\sin t\right)dt,$$

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Symbols:

${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/10.64.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.64, §10.64 and Ch.10

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