hyperbolic cosine function

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1: 4.35 Identities

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4.35.34 $$\sinh z=\sinh x\cos y+\mathrm{i}\cosh x\sin y,$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.5.49

Permalink:

http://dlmf.nist.gov/4.35.E34

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See also:

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

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4.35.35 $$\cosh z=\cosh x\cos y+\mathrm{i}\sinh x\sin y,$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.5.50

Permalink:

http://dlmf.nist.gov/4.35.E35

Encodings:

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See also:

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

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4.35.36 $$\tanh z=\frac{\sinh \left(2x\right)+\mathrm{i}\sin \left(2y\right)}{\cosh \left(2x\right)+\cos \left(2y\right)},$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.5.51

Permalink:

http://dlmf.nist.gov/4.35.E36

Encodings:

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See also:

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

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4.35.39 $$|\cosh z|={({\sinh }^{2}x+{\cos }^{2}y)}^{1/2}={\left(\frac{1}{2}(\cosh \left(2x\right)+\cos \left(2y\right))\right)}^{1/2},$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.5.56

Permalink:

http://dlmf.nist.gov/4.35.E39

Encodings:

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

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4.35.40 $$|\tanh z|={\left(\frac{\cosh \left(2x\right)-\cos \left(2y\right)}{\cosh \left(2x\right)+\cos \left(2y\right)}\right)}^{1/2}.$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.5.58

Permalink:

http://dlmf.nist.gov/4.35.E40

Encodings:

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See also:

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

2: 4.28 Definitions and Periodicity

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4.28.2 $$\cosh z=\frac{{\mathrm{e}}^z+{\mathrm{e}}^{-z}}{2},$$

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Defines:

$\cosh z$: hyperbolic cosine function

Symbols:

$\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

4.5.2

Permalink:

http://dlmf.nist.gov/4.28.E2

Encodings:

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See also:

Annotations for §4.28 and Ch.4

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4.28.3 $$\cosh z\pm \sinh z={\mathrm{e}}^{\pm z},$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.19 4.5.20

Permalink:

http://dlmf.nist.gov/4.28.E3

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See also:

Annotations for §4.28 and Ch.4

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4.28.4 $$\tanh z=\frac{\sinh z}{\cosh z},$$

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Defines:

$\tanh z$: hyperbolic tangent function

Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.3

Permalink:

http://dlmf.nist.gov/4.28.E4

Encodings:

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See also:

Annotations for §4.28 and Ch.4

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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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4.28.9 $$\cos \left(\mathrm{i}z\right)=\cosh z,$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.5.8

Permalink:

http://dlmf.nist.gov/4.28.E9

Encodings:

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See also:

Annotations for §4.28, §4.28 and Ch.4

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3: 4.32 Inequalities

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4.32.1 $$\cosh x\le {\left(\frac{\sinh x}{x}\right)}^3,$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.32 and Ch.4

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4.32.2 $x>0$,

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Symbols:

$\cos z$: cosine function, $\tanh z$: hyperbolic tangent function, $\sin z$: sine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.32 and Ch.4

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4.32.3 $$|\cosh x-\cosh y|\ge |x-y|\sqrt{\sinh x\sinh y},$$ $x>0$, $y>0$,

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Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.32 and Ch.4

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4: 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. …

5: 4.31 Special Values and Limits

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4.31.3 $$\underset{z\to 0}{lim}\frac{\cosh z-1}{z^2}=\frac{1}{2}.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/4.31.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.31 and Ch.4

6: 4.21 Identities

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4.21.37 $$\sin z=\sin x\cosh y+\mathrm{i}\cos x\sinh y,$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.3.55

Permalink:

http://dlmf.nist.gov/4.21.E37

Encodings:

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See also:

Annotations for §4.21(iv), §4.21 and Ch.4

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4.21.38 $$\cos z=\cos x\cosh y-\mathrm{i}\sin x\sinh y,$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.3.56

Permalink:

http://dlmf.nist.gov/4.21.E38

Encodings:

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See also:

Annotations for §4.21(iv), §4.21 and Ch.4

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4.21.39 $$\tan z=\frac{\sin \left(2x\right)+\mathrm{i}\sinh \left(2y\right)}{\cos \left(2x\right)+\cosh \left(2y\right)},$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $\tan z$: tangent function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.3.57

Permalink:

http://dlmf.nist.gov/4.21.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(iv), §4.21 and Ch.4

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4.21.41 $$|\sin z|={({\sin }^{2}x+{\sinh }^{2}y)}^{1/2}={\left(\frac{1}{2}\left(\cosh \left(2y\right)-\cos \left(2x\right)\right)\right)}^{1/2},$$

ⓘ

Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\sin z$: sine function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.3.59

Permalink:

http://dlmf.nist.gov/4.21.E41

Encodings:

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See also:

Annotations for §4.21(iv), §4.21 and Ch.4

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4.21.42 $$|\cos z|={({\cos }^{2}x+{\sinh }^{2}y)}^{1/2}={\left(\frac{1}{2}(\cosh \left(2y\right)+\cos \left(2x\right))\right)}^{1/2},$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.3.61

Permalink:

http://dlmf.nist.gov/4.21.E42

Encodings:

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See also:

Annotations for §4.21(iv), §4.21 and Ch.4

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7: 19.10 Relations to Other Functions

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19.10.2 $$(\sinh \varphi )R_C(1,{\cosh }^2\varphi )=\mathrm{gd}\left(\varphi \right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{gd}x$: Gudermannian function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $\varphi$: real or complex argument

Referenced by:

§19.10(ii), §19.6(ii)

Permalink:

http://dlmf.nist.gov/19.10.E2

Encodings:

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See also:

Annotations for §19.10(ii), §19.10 and Ch.19

8: 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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9: 4.18 Inequalities

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4.18.5 $$|\sinh y|\le |\sin z|\le \cosh y,$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\sin z$: sine function, $y$: real variable and $z$: complex variable

A&S Ref:

4.3.83

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.18, §4.18 and Ch.4

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4.18.6 $$|\sinh y|\le |\cos z|\le \cosh y,$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $y$: real variable and $z$: complex variable

A&S Ref:

4.3.84

Permalink:

http://dlmf.nist.gov/4.18.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.18, §4.18 and Ch.4

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4.18.8 $$|\cos z|\le \cosh |z|,$$

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Symbols:

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

A&S Ref:

4.3.86

Permalink:

http://dlmf.nist.gov/4.18.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.18, §4.18 and Ch.4

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10: 28.23 Expansions in Series of Bessel Functions

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28.23.2 $${\mathrm{me}}_{\nu }(0,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n{c}_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients and ${𝒞}_{\mu }^{(j)}$: cylinder functions

Permalink:

http://dlmf.nist.gov/28.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.3 $${\mathrm{me}}_{\nu }^{\prime }(0,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)=\mathrm{i}\tanh z\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n(\nu +2n)c_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients and ${𝒞}_{\mu }^{(j)}$: cylinder functions

Permalink:

http://dlmf.nist.gov/28.23.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.6 $${\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m}(0,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}{A}_{2\mathrm{\ell }}^{2m}(h^2){𝒞}_{2\mathrm{\ell }}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

A&S Ref:

20.6.12

Referenced by:

§28.23

Permalink:

http://dlmf.nist.gov/28.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.8 $${\mathrm{Mc}}_{2m+1}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m+1}(0,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}{A}_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.10 $${\mathrm{Ms}}_{2m+1}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{se}}_{2m+1}^{\prime }(0,h^2)\right)}^{-1}\tanh z\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}(2\mathrm{\ell }+1)B_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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