hyperbolic sine function

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11: 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:

pMML, png, TeX

See also:

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

12: 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:

pMML, png, TeX

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:

pMML, png, TeX

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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4.21.40 $$\cot z=\frac{\sin \left(2x\right)-\mathrm{i}\sinh \left(2y\right)}{\cosh \left(2y\right)-\cos \left(2x\right)}.$$

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Symbols:

$\cos z$: cosine function, $\cot z$: cotangent 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:

4.3.58

Permalink:

http://dlmf.nist.gov/4.21.E40

Encodings:

pMML, png, TeX

See also:

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

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

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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:

pMML, png, TeX

See also:

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

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13: 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

14: 4.37 Inverse Hyperbolic Functions

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4.37.4 $$\mathrm{Arccsch}z=\mathrm{Arcsinh}\left(1/z\right),$$

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Defines:

$\mathrm{Arccsch}z$: general inverse hyperbolic cosecant function

Symbols:

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

Permalink:

http://dlmf.nist.gov/4.37.E4

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. … ►

4.37.7 $$\mathrm{arccsch}z=\mathrm{arcsinh}\left(1/z\right),$$

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Defines:

$\mathrm{arccsch}z$: inverse hyperbolic cosecant function

Symbols:

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

A&S Ref:

4.6.4

Referenced by:

§4.37(iv), §4.45(i), §4.45(ii)

Permalink:

http://dlmf.nist.gov/4.37.E7

Encodings:

pMML, png, TeX

See also:

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

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4.37.10 $$\mathrm{arcsinh}\left(-z\right)=-\mathrm{arcsinh}z.$$

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Symbols:

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

A&S Ref:

4.6.11

Permalink:

http://dlmf.nist.gov/4.37.E10

Encodings:

pMML, png, TeX

See also:

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

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4.37.26 $$z=\sinh w,$$

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Symbols:

$\sinh z$: hyperbolic sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E26

Encodings:

pMML, png, TeX

See also:

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

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15: 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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4.18.9 $$|\sin z|\le \sinh |z|,$$

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Symbols:

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

A&S Ref:

4.3.87

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.18, §4.18 and Ch.4

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16: 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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	$\sinh \theta =a$	$\cosh \theta =a$	$\tanh \theta =a$	$\mathrm{csch}\theta =a$	$\mathrm{sech}\theta =a$	$\coth \theta =a$
$\sinh \theta$	$a$	${(a^2-1)}^{1/2}$	$a{(1-a^2)}^{-1/2}$	$a^{-1}$	$a^{-1}{(1-a^2)}^{1/2}$	${(a^2-1)}^{-1/2}$
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17: 28.23 Expansions in Series of Bessel Functions

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28.23.5 $${\mathrm{me}}_{\nu }^{\prime }(\frac{1}{2}\pi ,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)=\mathrm{i}{\mathrm{e}}^{\mathrm{i}\nu \pi /2}\coth z\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}(\nu +2n)c_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\sinh z),$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine 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.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.7 $${\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m}(\frac{1}{2}\pi ,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{A}_{2\mathrm{\ell }}^{2m}(h^2){𝒞}_{2\mathrm{\ell }}^{(j)}(2h\sinh z),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine 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.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.9 $${\mathrm{Mc}}_{2m+1}^{(j)}(z,h)={(-1)}^{m+1}{\left({\mathrm{ce}}_{2m+1}^{\prime }(\frac{1}{2}\pi ,h^2)\right)}^{-1}\coth z\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}(2\mathrm{\ell }+1)A_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\sinh z),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine 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.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.11 $${\mathrm{Ms}}_{2m+1}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{se}}_{2m+1}(\frac{1}{2}\pi ,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{B}_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\sinh z),$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine 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

A&S Ref:

20.6.13

Permalink:

http://dlmf.nist.gov/28.23.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.13 $${\mathrm{Ms}}_{2m+2}^{(j)}(z,h)={(-1)}^{m+1}{\left({\mathrm{se}}_{2m+2}^{\prime }(\frac{1}{2}\pi ,h^2)\right)}^{-1}\coth z\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}(2\mathrm{\ell }+2)B_{2\mathrm{\ell }+2}^{2m+2}(h^2){𝒞}_{2\mathrm{\ell }+2}^{(j)}(2h\sinh z).$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine 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

Referenced by:

§28.23

Permalink:

http://dlmf.nist.gov/28.23.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

…

18: 28.32 Mathematical Applications

§28.32 Mathematical Applications

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§28.32(i) Elliptical Coordinates and an Integral Relationship

… ► … ► … ►

19: 13.24 Series

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13.24.3 $$\exp \left(-\frac{1}{2}z\left(\coth t-\frac{1}{t}\right)\right){\left(\frac{t}{\sinh t}\right)}^{1-2\mu }=\sum _{s=0}^{\mathrm{\infty }}{p}_s^{(\mu )}(z){\left(-\frac{t}{z}\right)}^s.$$

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Symbols:

$\exp z$: exponential function, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $s$: nonnegative integer, $z$: complex variable and $p_s$: coefficents

Permalink:

http://dlmf.nist.gov/13.24.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.24(ii), §13.24 and Ch.13

…

20: 10.35 Generating Function and Associated Series

§10.35 Generating Function and Associated Series

… ►Jacobi–Anger expansions: for $z,\theta \in ℂ$, … ►

10.35.3 $${\mathrm{e}}^{z\sin \theta }=I_0\left(z\right)+2\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{I}_{2k+1}\left(z\right)\sin \left((2k+1)\theta \right)+2\sum _{k=1}^{\mathrm{\infty }}{(-1)}^k{I}_{2k}\left(z\right)\cos \left(2k\theta \right).$$

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Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\sin z$: sine function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.35

Referenced by:

§10.35

Permalink:

http://dlmf.nist.gov/10.35.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.35 and Ch.10

… ►

$$\cosh z=I_0\left(z\right)+2I_2\left(z\right)+2I_4\left(z\right)+2I_6\left(z\right)+\mathrm{\dots },$$

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$$\sinh z=2I_1\left(z\right)+2I_3\left(z\right)+2I_5\left(z\right)+\mathrm{\dots }.$$