hyperbolic trigonometric functions

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21: 22.5 Special Values

… ►In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. …

22: 4.23 Inverse Trigonometric Functions

… ►

4.23.39 $$\mathrm{gd}\left(x\right)={\int }_0^x\mathrm{sech}tdt,$$ .

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Defines:

$\mathrm{gd}x$: Gudermannian function

Symbols:

$dx$: differential of $x$ , $\mathrm{sech}z$: hyperbolic secant function, $\int$: integral and $x$: real variable

Referenced by:

§4.40(ii)

Permalink:

http://dlmf.nist.gov/4.23.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(viii), §4.23 and Ch.4

… ►

4.23.40 $$\mathrm{gd}\left(x\right)=2\arctan \left({\mathrm{e}}^x\right)-\frac{1}{2}\pi =\arcsin \left(\tanh x\right)=\mathrm{arccsc}\left(\coth x\right)=\arccos \left(\mathrm{sech}x\right)=\mathrm{arcsec}\left(\cosh x\right)=\arctan \left(\sinh x\right)=\mathrm{arccot}\left(\mathrm{csch}x\right).$$

ⓘ

Symbols:

$\mathrm{gd}x$: Gudermannian function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\coth z$: hyperbolic cotangent function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $\mathrm{arccsc}z$: arccosecant function, $\arccos z$: arccosine function, $\mathrm{arccot}z$: arccotangent function, $\mathrm{arcsec}z$: arcsecant function, $\arcsin z$: arcsine function, $\arctan z$: arctangent function and $x$: real variable

Referenced by:

§4.23(viii), §4.40(ii)

Permalink:

http://dlmf.nist.gov/4.23.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(viii), §4.23 and Ch.4

… ►

4.23.42 $${\mathrm{gd}}^{-1}\left(x\right)=\ln \tan \left(\frac{1}{2}x+\frac{1}{4}\pi \right)=\ln \left(\sec x+\tan x\right)=\mathrm{arcsinh}\left(\tan x\right)=\mathrm{arccsch}\left(\cot x\right)=\mathrm{arccosh}\left(\sec x\right)=\mathrm{arcsech}\left(\cos x\right)=\mathrm{arctanh}\left(\sin x\right)=\mathrm{arccoth}\left(\csc x\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $\cot z$: cotangent function, $\mathrm{arccsch}z$: inverse hyperbolic cosecant function, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{arccoth}z$: inverse hyperbolic cotangent function, $\mathrm{arcsech}z$: inverse hyperbolic secant function, $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\ln z$: principal branch of logarithm function, $\sec z$: secant function, $\sin z$: sine function, $\tan z$: tangent function and $x$: real variable

Referenced by:

§19.9(ii), §4.23(viii), §4.26(ii)

Permalink:

http://dlmf.nist.gov/4.23.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(viii), §4.23 and Ch.4

23: 14.5 Special Values

… ►

14.5.15 $$P_{\nu }^{1/2}\left(\cosh \xi \right)={\left(\frac{2}{\pi \sinh \xi }\right)}^{1/2}\cosh \left(\left(\nu +\frac{1}{2}\right)\xi \right),$$

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\nu$: general degree and $\xi >0$: variable

A&S Ref:

8.6.8 (modified)

Permalink:

http://dlmf.nist.gov/14.5.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(iii), §14.5 and Ch.14

►

14.5.16 $$P_{\nu }^{-1/2}\left(\cosh \xi \right)={\left(\frac{2}{\pi \sinh \xi }\right)}^{1/2}\frac{\sinh \left(\left(\nu +\frac{1}{2}\right)\xi \right)}{\nu +\frac{1}{2}},$$

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\nu$: general degree and $\xi >0$: variable

A&S Ref:

8.6.9 (modified)

Permalink:

http://dlmf.nist.gov/14.5.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(iii), §14.5 and Ch.14

►

14.5.17 $${𝑸}_{\nu }^{\pm 1/2}\left(\cosh \xi \right)={\left(\frac{\pi }{2\sinh \xi }\right)}^{1/2}\frac{\exp \left(-\left(\nu +\frac{1}{2}\right)\xi \right)}{\mathrm{\Gamma }\left(\nu +\frac{3}{2}\right)}.$$

ⓘ

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, $\exp z$: exponential function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\nu$: general degree and $\xi >0$: variable

A&S Ref:

8.6.10 (corrected) 8.6.11 (corrected)

Permalink:

http://dlmf.nist.gov/14.5.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(iii), §14.5 and Ch.14

… ►

14.5.19 $$P_{\nu }^{-\nu }\left(\cosh \xi \right)=\frac{{(\sinh \xi )}^{\nu }}{2^{\nu }\mathrm{\Gamma }\left(\nu +1\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\nu$: general degree and $\xi >0$: variable

A&S Ref:

8.6.16 (modified)

Permalink:

http://dlmf.nist.gov/14.5.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(iv), §14.5 and Ch.14

… ►

14.5.25 $$P_{-\frac{1}{2}}\left(\cosh \xi \right)=\frac{2}{\pi \cosh \left(\frac{1}{2}\xi \right)}K\left(\tanh \left(\frac{1}{2}\xi \right)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind and $\xi >0$: variable

A&S Ref:

8.13.2

Permalink:

http://dlmf.nist.gov/14.5.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(v), §14.5 and Ch.14

…

24: 4.43 Cubic Equations

§4.43 Cubic Equations

… ►

4.43.2 $$z^3+pz+q=0$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $a$: real or complex constant, $z$: complex variable, $p$: real constant and $q$: real constant

Referenced by:

(4.43.1), §4.43

Permalink:

http://dlmf.nist.gov/4.43.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

Cases (a), (b), and (c) of of this equation have been corrected. Formerly, with constants $C$ and $D$ given in Eq. (4.43.1), they read

1. $A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=C$, when and $C\le 1$.

2. $A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=C$, when and $C>1$.

3. $B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=D$, when $p>0$.

Reported 2014-10-31 by Masataka Urago

See also:

Annotations for §4.43 and Ch.4

…

25: 5.4 Special Values and Extrema

… ►

5.4.3 $$|\mathrm{\Gamma }\left(\mathrm{i}y\right)|={\left(\frac{\pi }{y\sinh \left(\pi y\right)}\right)}^{1/2},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.1.29

Referenced by:

§10.24, §5.4(i)

Permalink:

http://dlmf.nist.gov/5.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

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5.4.5 $$\mathrm{\Gamma }\left(\frac{1}{4}+\mathrm{i}y\right)\mathrm{\Gamma }\left(\frac{3}{4}-\mathrm{i}y\right)=\frac{\pi \sqrt{2}}{\cosh \left(\pi y\right)+\mathrm{i}\sinh \left(\pi y\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.1.32

Permalink:

http://dlmf.nist.gov/5.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

… ►

5.4.16 $$\mathrm{\Im }\psi \left(\mathrm{i}y\right)=\frac{1}{2y}+\frac{\pi }{2}\coth \left(\pi y\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\coth z$: hyperbolic cotangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.3.11

Permalink:

http://dlmf.nist.gov/5.4.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(ii), §5.4 and Ch.5

►

5.4.17 $$\mathrm{\Im }\psi \left(\frac{1}{2}+\mathrm{i}y\right)=\frac{\pi }{2}\tanh \left(\pi y\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\tanh z$: hyperbolic tangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.3.12

Permalink:

http://dlmf.nist.gov/5.4.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(ii), §5.4 and Ch.5

►

5.4.18 $$\mathrm{\Im }\psi \left(1+\mathrm{i}y\right)=-\frac{1}{2y}+\frac{\pi }{2}\coth \left(\pi y\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\coth z$: hyperbolic cotangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.3.13

Referenced by:

§5.4(ii)

Permalink:

http://dlmf.nist.gov/5.4.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(ii), §5.4 and Ch.5

…

26: 14.18 Sums

… ►

14.18.4 $$P_{\nu }\left(\cosh {\xi }_1\cosh {\xi }_2-\sinh {\xi }_1\sinh {\xi }_2\cos \varphi \right)=P_{\nu }\left(\cosh {\xi }_1\right)P_{\nu }\left(\cosh {\xi }_2\right)+2\sum _{m=1}^{\mathrm{\infty }}{(-1)}^m{P}_{\nu }^{-m}\left(\cosh {\xi }_1\right)P_{\nu }^m\left(\cosh {\xi }_2\right)\cos \left(m\varphi \right),$$

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $m$: nonnegative integer, $\nu$: general degree, $\varphi$: real and ${\xi }_1$, ${\xi }_2$: positive

Referenced by:

§14.18(ii)

Permalink:

http://dlmf.nist.gov/14.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.18(ii), §14.18 and Ch.14

►

14.18.5 $$Q_{\nu }\left(\cosh {\xi }_1\cosh {\xi }_2-\sinh {\xi }_1\sinh {\xi }_2\cos \varphi \right)=P_{\nu }\left(\cosh {\xi }_1\right)Q_{\nu }\left(\cosh {\xi }_2\right)+2\sum _{m=1}^{\mathrm{\infty }}{(-1)}^m{P}_{\nu }^{-m}\left(\cosh {\xi }_1\right)Q_{\nu }^m\left(\cosh {\xi }_2\right)\cos \left(m\varphi \right).$$

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $m$: nonnegative integer, $\nu$: general degree, $\varphi$: real and ${\xi }_1$, ${\xi }_2$: positive

Referenced by:

§14.18(ii)

Permalink:

http://dlmf.nist.gov/14.18.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.18(ii), §14.18 and Ch.14

…

27: 28.26 Asymptotic Approximations for Large $q$

… ►

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

ⓘ

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

►

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

ⓘ

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

… ►

28.26.3 $$\varphi =2h\sinh z-\left(m+\frac{1}{2}\right)\arctan \left(\sinh z\right).$$

ⓘ

Symbols:

$\sinh z$: hyperbolic sine function, $\arctan z$: arctangent function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.8 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E3

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

►

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

ⓘ

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

…

28: 20.10 Integrals

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20.10.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_1\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_2\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=-\frac{\mathrm{\ell }}{\sqrt{s}}\sinh \left(\beta \sqrt{s}\right)\mathrm{sech}\left(\mathrm{\ell }\sqrt{s}\right),$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

►

20.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_3\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_4\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=\frac{\mathrm{\ell }}{\sqrt{s}}\cosh \left(\beta \sqrt{s}\right)\mathrm{csch}\left(\mathrm{\ell }\sqrt{s}\right).$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

…

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

ⓘ

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

… ►

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

►

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

30: 4.36 Infinite Products and Partial Fractions

… ►

4.36.1 $$\sinh z=z\prod _{n=1}^{\mathrm{\infty }}\left(1+\frac{z^2}{n^2{\pi }^2}\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, $n$: integer and $z$: complex variable

A&S Ref:

4.5.68

Permalink:

http://dlmf.nist.gov/4.36.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.36 and Ch.4

►

4.36.2 $$\cosh z=\prod _{n=1}^{\mathrm{\infty }}\left(1+\frac{4z^2}{{(2n-1)}^2{\pi }^2}\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $n$: integer and $z$: complex variable

A&S Ref:

4.5.69

Permalink:

http://dlmf.nist.gov/4.36.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.36 and Ch.4

… ►

4.36.3 $$\coth z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{1}{z^2+n^2{\pi }^2},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\coth z$: hyperbolic cotangent function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.36.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.36 and Ch.4

►

4.36.4 $${\mathrm{csch}}^{2}z=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{{(z-n\pi \mathrm{i})}^2},$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/4.36.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.36 and Ch.4

►

4.36.5 $$\mathrm{csch}z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{z^2+n^2{\pi }^2}.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{csch}z$: hyperbolic cosecant function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.36.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.36 and Ch.4