hyperbolic cosine function

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21: 14.18 Sums

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

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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

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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

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22: 4.33 Maclaurin Series and Laurent Series

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4.33.2 $$\cosh z=1+\frac{z^2}{2!}+\frac{z^4}{4!}+\mathrm{\cdots }.$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function and $z$: complex variable

A&S Ref:

4.5.63

Permalink:

http://dlmf.nist.gov/4.33.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.33 and Ch.4

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23: 29.5 Special Cases and Limiting Forms

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29.5.5 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{\mathrm{𝐸𝑐}}_{\nu }^m(0,k^2)}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(0,k^2)}=\frac{1}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu ,\frac{1}{2}\mu +\frac{1}{2}\nu +\frac{1}{2}}{\frac{1}{2}};{\tanh }^{2}z),$$ $m$ even,

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.5 and Ch.29

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29.5.6 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{d{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)/dz|}_{z=0}}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{d{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)/dz|}_{z=0}}=\frac{\tanh z}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu +\frac{1}{2},\frac{1}{2}\mu +\frac{1}{2}\nu +1}{\frac{3}{2}};{\tanh }^{2}z),$$ $m$ odd,

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.5 and Ch.29

…

24: 14.5 Special Values

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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

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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

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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

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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

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25: 24.7 Integral Representations

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24.7.7 $$B_{2n}\left(x\right)={(-1)}^{n+1}2n{\int }_0^{\mathrm{\infty }}\frac{\cos \left(2\pi x\right)-{\mathrm{e}}^{-2\pi t}}{\cosh \left(2\pi t\right)-\cos \left(2\pi x\right)}{t}^{2n-1}dt,$$ $n=1,2,\mathrm{\dots }$,

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\int$: integral, $n$: integer, $t$: real or complex and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(ii), §24.7 and Ch.24

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24.7.8 $$B_{2n+1}\left(x\right)={(-1)}^{n+1}(2n+1){\int }_0^{\mathrm{\infty }}\frac{\sin \left(2\pi x\right)}{\cosh \left(2\pi t\right)-\cos \left(2\pi x\right)}{t}^{2n}dt.$$

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: 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, $t$: real or complex and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(ii), §24.7 and Ch.24

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24.7.9 $$E_{2n}\left(x\right)={(-1)}^{n}4{\int }_0^{\mathrm{\infty }}\frac{\sin \left(\pi x\right)\cosh \left(\pi t\right)}{\cosh \left(2\pi t\right)-\cos \left(2\pi x\right)}{t}^{2n}dt,$$

ⓘ

Symbols:

$E_n\left(x\right)$: Euler polynomials, $\pi$: 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, $t$: real or complex and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(ii), §24.7 and Ch.24

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24.7.10 $$E_{2n+1}\left(x\right)={(-1)}^{n+1}4{\int }_0^{\mathrm{\infty }}\frac{\cos \left(\pi x\right)\sinh \left(\pi t\right)}{\cosh \left(2\pi t\right)-\cos \left(2\pi x\right)}{t}^{2n+1}dt.$$

ⓘ

Symbols:

$E_n\left(x\right)$: Euler polynomials, $\pi$: 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, $n$: integer, $t$: real or complex and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(ii), §24.7 and Ch.24

…

26: 28.28 Integrals, Integral Representations, and Integral Equations

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28.28.1 $$w=\cosh z\cos t\cos \alpha +\sinh z\sin t\sin \alpha .$$

ⓘ

Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\sin z$: sine function, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter

Permalink:

http://dlmf.nist.gov/28.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

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28.28.10

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\mathrm{ph}$: phase, $h$: parameter and $z$: complex variable

Permalink:

http://dlmf.nist.gov/28.28.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

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28.28.37 $$\frac{\cosh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\sin t{\mathrm{se}}_n^{\prime }(t,h^2){\mathrm{se}}_m(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{p+1}\mathrm{i}h{\widehat{\alpha }}_{n,m}^{(s)}{\mathrm{Ds}}_1(n,m,z),$$

ⓘ

Symbols:

${\mathrm{Ds}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{se}}_n(z,q)$: Mathieu 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, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $n$: integer, $z$: complex variable and ${\widehat{\alpha }}_{n,m}$

Permalink:

http://dlmf.nist.gov/28.28.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iv), §28.28 and Ch.28

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28.28.41 $$\frac{\cosh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\sin t{\mathrm{se}}_n(t,h^2){\mathrm{ce}}_m(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{p+1}\mathrm{i}h{\widehat{\beta }}_{n,m}{\mathrm{Dsc}}_0(n,m,z),$$

ⓘ

Symbols:

${\mathrm{Dsc}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{se}}_n(z,q)$: Mathieu 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, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $n$: integer, $z$: complex variable and ${\widehat{\beta }}_{n,m}$

Permalink:

http://dlmf.nist.gov/28.28.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iv), §28.28 and Ch.28

… ►

28.28.48 $$\frac{\cosh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\sin t{\mathrm{ce}}_n^{\prime }(t,h^2){\mathrm{ce}}_m(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{p+1}\mathrm{i}h{\widehat{\alpha }}_{n,m}^{(c)}{\mathrm{Dc}}_1(n,m,z),$$

ⓘ

Symbols:

${\mathrm{Dc}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{ce}}_n(z,q)$: Mathieu 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, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $n$: integer, $z$: complex variable and ${\widehat{\alpha }}_{n,m}$

Permalink:

http://dlmf.nist.gov/28.28.E48

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iv), §28.28 and Ch.28

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27: 14.12 Integral Representations

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14.12.3 $${𝖰}_{\nu }^{\mu }\left(\cos \theta \right)=\frac{{\pi }^{1/2}\mathrm{\Gamma }\left(\nu +\mu +1\right){(\sin \theta )}^{\mu }}{2^{\mu +1}\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}\left({\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{(\cos \theta +\mathrm{i}\sin \theta \cosh t)}^{\nu +\mu +1}}dt+{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{(\cos \theta -\mathrm{i}\sin \theta \cosh t)}^{\nu +\mu +1}}dt\right),$$ , $\mathrm{\Re }\mu >-\frac{1}{2}$, $\mathrm{\Re }\nu \pm \mu >-1$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second 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, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(i), §14.12(i), §14.12 and Ch.14

… ►

14.12.4 $$P_{\nu }^{-\mu }\left(x\right)=\frac{2^{1/2}\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right){\left(x^2-1\right)}^{\mu /2}}{{\pi }^{1/2}\mathrm{\Gamma }\left(\nu +\mu +1\right)\mathrm{\Gamma }\left(\mu -\nu \right)}{\int }_0^{\mathrm{\infty }}\frac{\cosh \left(\left(\nu +\frac{1}{2}\right)t\right)}{{(x+\cosh t)}^{\mu +(1/2)}}dt,$$ $\nu +\mu \ne -1,-2,-3,\mathrm{\dots }$, $\mathrm{\Re }\left(\mu -\nu \right)>0$.

ⓘ

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, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(ii), §14.12 and Ch.14

… ►

14.12.6 $${𝑸}_{\nu }^{\mu }\left(x\right)=\frac{{\pi }^{1/2}{\left(x^2-1\right)}^{\mu /2}}{2^{\mu }\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{\left(x+{(x^2-1)}^{1/2}\cosh t\right)}^{\nu +\mu +1}}dt,$$ $\mathrm{\Re }\left(\nu +1\right)>\mathrm{\Re }\mu >-\frac{1}{2}$.

ⓘ

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, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.8.2 (modified)

Permalink:

http://dlmf.nist.gov/14.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(ii), §14.12 and Ch.14

… ►

14.12.9 $${𝑸}_n^m\left(x\right)=\frac{1}{n!}{\int }_0^u{\left(x-{\left(x^2-1\right)}^{1/2}\cosh t\right)}^n\cosh \left(mt\right)dt,$$

ⓘ

Symbols:

${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $u$

Permalink:

http://dlmf.nist.gov/14.12.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(ii), §14.12 and Ch.14

… ►

14.12.14 $${𝑸}_n\left(x\right)=\frac{1}{n!}{\int }_0^{\mathrm{\infty }}\frac{dt}{{\left(x+{(x^2-1)}^{1/2}\cosh t\right)}^{n+1}}.$$

ⓘ

Symbols:

$dx$: differential of $x$ , $!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $\int$: integral, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/14.12.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(ii), §14.12(ii), §14.12 and Ch.14

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28: 4.36 Infinite Products and Partial Fractions

… ►

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

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29: 20.10 Integrals

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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

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30: 10.35 Generating Function and Associated Series

§10.35 Generating Function and Associated Series

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

10.35.2 $${\mathrm{e}}^{z\cos \theta }=I_0\left(z\right)+2\sum _{k=1}^{\mathrm{\infty }}{I}_k\left(z\right)\cos \left(k\theta \right),$$

ⓘ

Symbols:

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

A&S Ref:

9.6.34

Referenced by:

§10.35

Permalink:

http://dlmf.nist.gov/10.35.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.35 and Ch.10

►

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:

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See also:

Annotations for §10.35 and Ch.10

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

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