circular trigonometric functions

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1: 4.1 Special Notation

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$k,m,n$	integers.
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… ►The main functions treated in this chapter are the logarithm $\ln z$, $\mathrm{Ln}z$; the exponential $\exp z$, ${\mathrm{e}}^z$; the circular trigonometric (or just trigonometric) functions $\sin z$, $\cos z$, $\tan z$, $\csc z$, $\sec z$, $\cot z$; the inverse trigonometric functions $\arcsin z$, $\mathrm{Arcsin}z$, etc. …

2: 4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

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$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
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3: 4.21 Identities

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4.21.1 $$\sin u\pm \cos u=\sqrt{2}\sin \left(u\pm \frac{1}{4}\pi \right)=\pm \sqrt{2}\cos \left(u\mp \frac{1}{4}\pi \right).$$

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Symbols:

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

Referenced by:

§4.21(i), Erratum (V1.0.7) for Equation (4.21.1)

Permalink:

http://dlmf.nist.gov/4.21.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.0.7):

Originally the symbol $\pm$ was missing after the second equal sign.

Suggested 2012-09-27 by Dennis M. Heim

See also:

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

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4.21.35 $$\sin \left(nz\right)=2^{n-1}\prod _{k=0}^{n-1}\sin \left(z+\frac{k\pi }{n}\right),$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

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

Referenced by:

§23.10(iii), §4.21(iii)

Permalink:

http://dlmf.nist.gov/4.21.E35

Encodings:

pMML, png, TeX

See also:

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

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4: 4.22 Infinite Products and Partial Fractions

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4.22.1 $$\sin z=z\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{z^2}{n^2{\pi }^2}\right),$$

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Symbols:

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

A&S Ref:

4.3.89

Permalink:

http://dlmf.nist.gov/4.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

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4.22.2 $$\cos z=\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{4z^2}{{(2n-1)}^2{\pi }^2}\right).$$

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Symbols:

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

A&S Ref:

4.3.90

Permalink:

http://dlmf.nist.gov/4.22.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

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4.22.3 $$\cot z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{1}{z^2-n^2{\pi }^2},$$

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Symbols:

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

A&S Ref:

4.3.91

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/4.22.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

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4.22.4 $${\csc }^{2}z=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{{(z-n\pi )}^2},$$

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Symbols:

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

A&S Ref:

4.3.92

Permalink:

http://dlmf.nist.gov/4.22.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

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4.22.5 $$\csc z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{z^2-n^2{\pi }^2}.$$

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Symbols:

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

A&S Ref:

4.3.93

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/4.22.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

5: 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

6: 7.5 Interrelations

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7.5.3 $$C\left(z\right)=\frac{1}{2}+\mathrm{f}\left(z\right)\sin \left(\frac{1}{2}\pi z^2\right)-\mathrm{g}\left(z\right)\cos \left(\frac{1}{2}\pi z^2\right),$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

7.3.9

Referenced by:

§7.12(ii), §7.5

Permalink:

http://dlmf.nist.gov/7.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.4 $$S\left(z\right)=\frac{1}{2}-\mathrm{f}\left(z\right)\cos \left(\frac{1}{2}\pi z^2\right)-\mathrm{g}\left(z\right)\sin \left(\frac{1}{2}\pi z^2\right).$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

7.3.10

Referenced by:

§7.12(ii), §7.5

Permalink:

http://dlmf.nist.gov/7.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.6 $${\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}{z}^2}(\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right))=\frac{1}{2}(1\pm \mathrm{i})-(C\left(z\right)\pm \mathrm{i}S\left(z\right)).$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.9 $$C\left(z\right)\pm \mathrm{i}S\left(z\right)=\frac{1}{2}(1\pm \mathrm{i})\left(1-{\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}{z}^2}w\left(\mathrm{i}\zeta \right)\right).$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.12 $${|ℱ\left(x\right)|}^2=2+{\mathrm{f}}^2\left(-x\right)+{\mathrm{g}}^2\left(-x\right)-2\sqrt{2}\cos \left(\frac{1}{4}\pi +\frac{1}{2}\pi x^2\right)\mathrm{f}\left(-x\right)-2\sqrt{2}\cos \left(\frac{1}{4}\pi -\frac{1}{2}\pi x^2\right)\mathrm{g}\left(-x\right),$$ $x\le 0$.

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $ℱ\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $x$: real variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7: 4.25 Continued Fractions

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4.25.1 $$\tan z=\frac{z}{1-}\frac{z^2}{3-}\frac{z^2}{5-}\frac{z^2}{7-}\mathrm{\cdots },$$ $z\ne \pm \frac{1}{2}\pi$, $\pm \frac{3}{2}\pi$, $\mathrm{\dots }$.

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Symbols:

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

A&S Ref:

4.3.94

Permalink:

http://dlmf.nist.gov/4.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

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4.25.2 $$\tan \left(az\right)=\frac{a\tan z}{1+}\frac{(1-a^2){\tan }^{2}z}{3+}\frac{(4-a^2){\tan }^{2}z}{5+}\frac{(9-a^2){\tan }^{2}z}{7+}\mathrm{\cdots },$$ , $az\ne \pm \frac{1}{2}\pi ,\pm \frac{3}{2}\pi ,\mathrm{\dots }$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $\tan z$: tangent function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.3.95

Permalink:

http://dlmf.nist.gov/4.25.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

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8: 4.14 Definitions and Periodicity

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4.14.8 $$\sin \left(z+2k\pi \right)=\sin z,$$

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Symbols:

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

A&S Ref:

4.3.7

Permalink:

http://dlmf.nist.gov/4.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

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4.14.9 $$\cos \left(z+2k\pi \right)=\cos z,$$

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Symbols:

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

A&S Ref:

4.3.8

Permalink:

http://dlmf.nist.gov/4.14.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

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4.14.10 $$\tan \left(z+k\pi \right)=\tan z.$$

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Symbols:

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

A&S Ref:

4.3.9

Permalink:

http://dlmf.nist.gov/4.14.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

9: 10.64 Integral Representations

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10.64.1 $${\mathrm{ber}}_n\left(x\sqrt{2}\right)=\frac{{(-1)}^n}{\pi }{\int }_0^{\pi }\cos \left(x\sin t-nt\right)\cosh \left(x\sin t\right)dt,$$

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Symbols:

${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: 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 and $x$: real variable

Permalink:

http://dlmf.nist.gov/10.64.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.64, §10.64 and Ch.10

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10.64.2 $${\mathrm{bei}}_n\left(x\sqrt{2}\right)=\frac{{(-1)}^n}{\pi }{\int }_0^{\pi }\sin \left(x\sin t-nt\right)\sinh \left(x\sin t\right)dt.$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/10.64.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.64, §10.64 and Ch.10

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10: 19.3 Graphics

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Figure 19.3.6: $\mathrm{\Pi }(\varphi ,2,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 3$, . …If ${\sin }^2\varphi =1$ ($>k^2$), then the function reduces to $\mathrm{\Pi }(2,k)$ with Cauchy principal value $K\left(k\right)-\mathrm{\Pi }(\frac{1}{2}{k}^2,k)$, which tends to $-\mathrm{\infty }$ as $k^2\to 1-$. … Magnify 3D Help

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