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21: 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

22: 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

23: 4.15 Graphics

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4.15.1 $$\cos \left(x+\mathrm{i}y\right)=\sin \left(x+\frac{1}{2}\pi +\mathrm{i}y\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.15(iii), §4.15 and Ch.4

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24: 4.16 Elementary Properties

§4.16 Elementary Properties

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Table 4.16.1: Signs of the trigonometric functions in the four quadrants.

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Quadrant	$\sin \theta ,\csc \theta$	$\cos \theta ,\sec \theta$	$\tan \theta ,\cot \theta$
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Table 4.16.2: Trigonometric functions: quarter periods and change of sign.

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$x$	$-\theta$	$\frac{1}{2}\pi \pm \theta$	$\pi \pm \theta$	$\frac{3}{2}\pi \pm \theta$	$2\pi \pm \theta$
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Table 4.16.3: Trigonometric functions: interrelations. …

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	$\sin \theta =a$	$\cos \theta =a$	$\tan \theta =a$	$\csc \theta =a$	$\sec \theta =a$	$\cot \theta =a$
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25: 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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26: 4.17 Special Values and Limits

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4.17.1 $$\underset{z\to 0}{lim}\frac{\sin z}{z}=1,$$

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Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.74

Permalink:

http://dlmf.nist.gov/4.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

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27: 28.28 Integrals, Integral Representations, and Integral Equations

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

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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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28.28.26 $$\frac{\cosh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\sin t{\mathrm{me}}_{\nu }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{m+1}\mathrm{i}h{\alpha }_{\nu ,m}^{(1)}{\mathrm{D}}_0(\nu ,\nu +2m+1,z),$$

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Symbols:

${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: 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, $z$: complex variable, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$

Permalink:

http://dlmf.nist.gov/28.28.E26

Encodings:

pMML, png, TeX

See also:

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

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28.28.29 $$\frac{\cosh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\sin t{\mathrm{me}}_{\nu }^{\prime }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{m+1}\mathrm{i}h{\alpha }_{\nu ,m}^{(0)}{\mathrm{D}}_1(\nu ,\nu +2m+1,z),$$

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Symbols:

${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: 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, $z$: complex variable, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$

Permalink:

http://dlmf.nist.gov/28.28.E29

Encodings:

pMML, png, TeX

See also:

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

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

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

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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

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28: 36.13 Kelvin’s Ship-Wave Pattern

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36.13.5 $$|\varphi |={\varphi }_c=\arcsin \left(\frac{1}{3}\right)={19}^{\circ }.47122.$$

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Symbols:

$\arcsin z$: arcsine function

Permalink:

http://dlmf.nist.gov/36.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.13 and Ch.36

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36.13.8 $$z(\rho ,\varphi )=2\pi \left({\rho }^{-1/3}u(\varphi )\cos \left(\rho \tilde{f}(\varphi )\right)\mathrm{Ai}\left(-{\rho }^{2/3}\mathrm{\Delta }(\varphi )\right)(1+O\left(1/\rho \right))+{\rho }^{-2/3}v(\varphi )\sin \left(\rho \tilde{f}(\varphi )\right){\mathrm{Ai}}^{\prime }\left(-{\rho }^{2/3}\mathrm{\Delta }(\varphi )\right)(1+O\left(1/\rho \right))\right),$$ $\rho \to \mathrm{\infty }$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z(\varphi ,\rho )$: wave height approximant, $u(\varphi )$: function and $v(\varphi )$: function

Referenced by:

Figure 36.13.1, Figure 36.13.1

Permalink:

http://dlmf.nist.gov/36.13.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.13 and Ch.36

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►►

Figure 36.13.1: Kelvin’s ship wave pattern, computed from the uniform asymptotic approximation (36.13.8), as a function of $x=\rho \cos \varphi$, $y=\rho \sin \varphi$. Magnify

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29: 28.32 Mathematical Applications

§28.32 Mathematical Applications

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§28.32(i) Elliptical Coordinates and an Integral Relationship

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30: 20.5 Infinite Products and Related Results

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20.5.1 $${\theta }_1(z,q)=2q^{1/4}\sin z\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1-2q^{2n}\cos \left(2z\right)+q^{4n}\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\sin z$: sine function, $n$: integer, $z$: complex and $q$: nome

Referenced by:

§20.4(i), §20.5(i), §23.17(iii)

Permalink:

http://dlmf.nist.gov/20.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5 and Ch.20

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20.5.10 $$\frac{{\theta }_1^{\prime }(z,q)}{{\theta }_1(z,q)}-\cot z=4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{1-2q^{2n}\cos \left(2z\right)+q^{4n}}=4\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{1-q^{2n}}\sin \left(2nz\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\cot z$: cotangent function, $\sin z$: sine function, $n$: integer, $z$: complex and $q$: nome

A&S Ref:

16.29.1

Referenced by:

§20.5(ii), §20.5(ii)

Permalink:

http://dlmf.nist.gov/20.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(ii), §20.5 and Ch.20

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20.5.11 $$\frac{{\theta }_2^{\prime }(z,q)}{{\theta }_2(z,q)}+\tan z=-4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n}}{1+2q^{2n}\cos \left(2z\right)+q^{4n}}=4\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{q^{2n}}{1-q^{2n}}\sin \left(2nz\right).$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\sin z$: sine function, $\tan z$: tangent function, $n$: integer, $z$: complex and $q$: nome

A&S Ref:

16.29.2

Referenced by:

§20.5(ii)

Permalink:

http://dlmf.nist.gov/20.5.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(ii), §20.5 and Ch.20

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20.5.12 $$\frac{{\theta }_3^{\prime }(z,q)}{{\theta }_3(z,q)}=-4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n-1}}{1+2q^{2n-1}\cos \left(2z\right)+q^{4n-2}}=4\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{q^n}{1-q^{2n}}\sin \left(2nz\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\sin z$: sine function, $n$: integer, $z$: complex and $q$: nome

A&S Ref:

16.29.3

Permalink:

http://dlmf.nist.gov/20.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(ii), §20.5 and Ch.20

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20.5.13 $$\frac{{\theta }_4^{\prime }(z,q)}{{\theta }_4(z,q)}=4\sin \left(2z\right)\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n-1}}{1-2q^{2n-1}\cos \left(2z\right)+q^{4n-2}}=4\sum _{n=1}^{\mathrm{\infty }}\frac{q^n}{1-q^{2n}}\sin \left(2nz\right).$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\sin z$: sine function, $n$: integer, $z$: complex and $q$: nome

A&S Ref:

16.29.4

Referenced by:

§20.5(ii)

Permalink:

http://dlmf.nist.gov/20.5.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(ii), §20.5 and Ch.20

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