circular trigonometric functions

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11: 4.18 Inequalities

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4.18.1 $$\frac{2x}{\pi }\le \sin x\le x,$$ $0\le x\le \frac{1}{2}\pi$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $x$: real variable

A&S Ref:

4.3.79 4.3.80

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.18, §4.18 and Ch.4

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4.18.2 $$x\le \tan x,$$ ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\tan z$: tangent function and $x$: real variable

A&S Ref:

4.3.80

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.18, §4.18 and Ch.4

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4.18.3 $$\cos x\le \frac{\sin x}{x}\le 1,$$ $0\le x\le \pi$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $x$: real variable

A&S Ref:

4.3.81

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.18, §4.18 and Ch.4

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4.18.4 .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $x$: real variable

A&S Ref:

4.3.82

Referenced by:

§4.18

Permalink:

http://dlmf.nist.gov/4.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.18, §4.18 and Ch.4

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12: 4.26 Integrals

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4.26.3 $$\int \tan xdx=-\ln \left(\cos x\right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $\tan z$: tangent function and $x$: real variable

A&S Ref:

4.3.115

Permalink:

http://dlmf.nist.gov/4.26.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

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4.26.4 $$\int \csc xdx=\ln \left(\tan \frac{1}{2}x\right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $\tan z$: tangent function and $x$: real variable

A&S Ref:

4.3.116

Permalink:

http://dlmf.nist.gov/4.26.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

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4.26.6 $$\int \cot xdx=\ln \left(\sin x\right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $\sin z$: sine function and $x$: real variable

A&S Ref:

4.3.118

Permalink:

http://dlmf.nist.gov/4.26.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

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4.26.10 $${\int }_0^{\pi }\cos \left(mt\right)\cos \left(nt\right)dt=0,$$ $m\ne n$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $m$: integer and $n$: integer

A&S Ref:

4.3.140

Permalink:

http://dlmf.nist.gov/4.26.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(iii), §4.26(iii), §4.26 and Ch.4

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4.26.11 $${\int }_0^{\pi }{\sin }^2\left(nt\right)dt={\int }_0^{\pi }{\cos }^2\left(nt\right)dt=\frac{1}{2}\pi ,$$ $n\ne 0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $n$: integer

A&S Ref:

4.3.141

Permalink:

http://dlmf.nist.gov/4.26.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(iii), §4.26(iii), §4.26 and Ch.4

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13: 19.2 Definitions

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19.2.16 $$\mathrm{el3}(x,k_c,p)={\int }_0^{\arctan x}\frac{d\theta }{({\cos }^2\theta +p{\sin }^2\theta )\sqrt{{\cos }^2\theta +k_c^2{\sin }^2\theta }}=\mathrm{\Pi }(\arctan x,1-p,k),$$ $x^2\ne -1/p$.

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Defines:

$\mathrm{el3}(x,k_c,p)$: Bulirsch’s incomplete elliptic integral of the third kind

Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\int$: integral, $\arctan z$: arctangent function, $\sin z$: sine function, $k$: real or complex modulus, $p$: real parameter not equal to zero, $k_c$: real or complex variable not equal to zero and $x$: real or complex variable

Permalink:

http://dlmf.nist.gov/19.2.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iii), §19.2 and Ch.19

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19.2.18 $$R_C(x,y)=\frac{1}{\sqrt{y-x}}\arctan \sqrt{\frac{y-x}{x}}=\frac{1}{\sqrt{y-x}}\arccos \sqrt{x/y},$$ ,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\arccos z$: arccosine function and $\arctan z$: arctangent function

Referenced by:

§19.2(iv)

Permalink:

http://dlmf.nist.gov/19.2.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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19.2.19 $$R_C(x,y)=\frac{1}{\sqrt{x-y}}\mathrm{arctanh}\sqrt{\frac{x-y}{x}}=\frac{1}{\sqrt{x-y}}\ln \frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{y}},$$ .

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{arctanh}z$: inverse hyperbolic tangent function and $\ln z$: principal branch of logarithm function

Referenced by:

§19.26(i), Figure 19.3.2, Figure 19.3.2, §19.36(ii)

Permalink:

http://dlmf.nist.gov/19.2.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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19.2.20 $$R_C(x,y)=\sqrt{\frac{x}{x-y}}{R}_C(x-y,-y)=\frac{1}{\sqrt{x-y}}\mathrm{arctanh}\sqrt{\frac{x}{x-y}}=\frac{1}{\sqrt{x-y}}\ln \frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{-y}},$$ .

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{arctanh}z$: inverse hyperbolic tangent function and $\ln z$: principal branch of logarithm function

Referenced by:

§19.2(iv), §19.20(iii), §19.25(i), Figure 19.3.2, Figure 19.3.2, §19.36(i)

Permalink:

http://dlmf.nist.gov/19.2.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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19.2.22 $$R_C(x,y)=\frac{2}{\pi }{\int }_0^{\pi /2}{R}_C(y,x{\cos }^2\theta +y{\sin }^2\theta )d\theta .$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $\sin z$: sine function

Referenced by:

§19.2(iv), §19.2(iv)

Permalink:

http://dlmf.nist.gov/19.2.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

14: 7.14 Integrals

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7.14.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}C\left(t\right)dt=\frac{1}{a}\mathrm{f}\left(\frac{a}{\pi }\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.27 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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7.14.6 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}S\left(t\right)dt=\frac{1}{a}\mathrm{g}\left(\frac{a}{\pi }\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.28 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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15: 19.11 Addition Theorems

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19.11.6_5 $$R_C(\gamma -\delta ,\gamma )=\frac{-1}{\sqrt{\delta }}\arctan \left(\frac{\sqrt{\delta }\sin \theta \sin \varphi \sin \psi }{{\alpha }^2-1-{\alpha }^2\cos \theta \cos \varphi \cos \psi }\right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $\arctan z$: arctangent function, $\sin z$: sine function, $\varphi$: real or complex argument, ${\alpha }^2$: real or complex parameter, $\theta$: amplitude, $\psi$: angle, $\gamma$ and $\delta$

Referenced by:

§19.11(i), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)

Permalink:

http://dlmf.nist.gov/19.11.E6_5

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.28):

This equation was given a number.

See also:

Annotations for §19.11(i), §19.11 and Ch.19

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16: 4.28 Definitions and Periodicity

… ►The functions $\sinh z$ and $\cosh z$ have period $2\pi \mathrm{i}$, and $\tanh z$ has period $\pi \mathrm{i}$. …

17: 4.36 Infinite Products and Partial Fractions

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

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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

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

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

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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

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

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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

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

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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

18: 7.4 Symmetry

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$\mathrm{f}\left(-z\right)=\sqrt{2}\cos \left(\frac{1}{4}\pi +\frac{1}{2}\pi z^2\right)-\mathrm{f}\left(z\right),$

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$\mathrm{g}\left(-z\right)=\sqrt{2}\sin \left(\frac{1}{4}\pi +\frac{1}{2}\pi z^2\right)-\mathrm{g}\left(z\right).$

19: 4.45 Methods of Computation

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4.45.11 $$\arctan x=\frac{\pi }{2}-\frac{1}{x}+\frac{1}{3x^3}-\frac{1}{5x^5}+\mathrm{\dots };$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function and $x$: real variable

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.45.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.45(i), §4.45(i), §4.45 and Ch.4

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20: 11.10 Anger–Weber Functions

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11.10.6 $$f(\nu ,z)=\frac{(z-\nu )}{\pi z^2}\sin \left(\pi \nu \right),$$ $w={𝐉}_{\nu }\left(z\right)$,

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(ii), §11.10 and Ch.11

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11.10.7 $$f(\nu ,z)=-\frac{1}{\pi z^2}(z+\nu +(z-\nu )\cos \left(\pi \nu \right)),$$ $w={𝐄}_{\nu }\left(z\right)$.

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Symbols:

${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(ii), §11.10 and Ch.11

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11.10.9 $${𝐄}_{\nu }\left(z\right)=\sin \left(\frac{1}{2}\pi \nu \right)S_1(\nu ,z)-\cos \left(\frac{1}{2}\pi \nu \right)S_2(\nu ,z),$$

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Symbols:

${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order

Referenced by:

(11.11.6)

Permalink:

http://dlmf.nist.gov/11.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(iii), §11.10 and Ch.11

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11.10.13 $$\sin \left(\pi \nu \right){𝐉}_{\nu }\left(z\right)=\cos \left(\pi \nu \right){𝐄}_{\nu }\left(z\right)-{𝐄}_{-\nu }\left(z\right),$$

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order

A&S Ref:

12.3.4

Referenced by:

§11.10(v)

Permalink:

http://dlmf.nist.gov/11.10.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(v), §11.10 and Ch.11

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11.10.18 $${𝐄}_{\nu }\left(z\right)=-\frac{1}{\pi }(1+\cos \left(\pi \nu \right))s_{0,\nu }\left(z\right)-\frac{\nu }{\pi }(1-\cos \left(\pi \nu \right))s_{-1,\nu }\left(z\right).$$

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Symbols:

$s_{\mu ,\nu }\left(z\right)$: Lommel function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.10(vi)

Permalink:

http://dlmf.nist.gov/11.10.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vi), §11.10 and Ch.11

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