4 Elementary FunctionsTrigonometric Functions4.13 Lambert $W$-Function4.15 Graphics

§4.14 Definitions and Periodicity

ⓘ

Keywords:

analytic properties, definitions, periodicity, trigonometric functions, zeros

Notes:

See Levinson and Redheffer (1970, pp. 55–57).

Referenced by:

§1.9(i)

Permalink:

http://dlmf.nist.gov/4.14

See also:

Annotations for Ch.4

4.14.1		$\sin z$	$={\displaystyle \frac{{\mathrm{e}}^{\mathrm{i}z}-{\mathrm{e}}^{-\mathrm{i}z}}{2\mathrm{i}}},$
ⓘ Defines: $\sin z$: sine function Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable A&S Ref: 4.3.1 Referenced by: §4.45(ii), (9.5.3) Permalink: http://dlmf.nist.gov/4.14.E1 Encodings: TeX, pMML, png See also: Annotations for §4.14 and Ch.4
4.14.2		$\cos z$	$={\displaystyle \frac{{\mathrm{e}}^{\mathrm{i}z}+{\mathrm{e}}^{-\mathrm{i}z}}{2}},$
ⓘ Defines: $\cos z$: cosine function Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable A&S Ref: 4.3.2 Referenced by: (9.5.3) Permalink: http://dlmf.nist.gov/4.14.E2 Encodings: TeX, pMML, png See also: Annotations for §4.14 and Ch.4
4.14.3		$\cos z\pm \mathrm{i}\sin z$	$={\mathrm{e}}^{\pm \mathrm{i}z},$
ⓘ Symbols: $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function and $z$: complex variable Permalink: http://dlmf.nist.gov/4.14.E3 Encodings: TeX, pMML, png See also: Annotations for §4.14 and Ch.4
4.14.4		$\tan z$	$={\displaystyle \frac{\sin z}{\cos z}},$
ⓘ Defines: $\tan z$: tangent function Symbols: $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable A&S Ref: 4.3.3 Referenced by: §4.14, §4.45(i) Permalink: http://dlmf.nist.gov/4.14.E4 Encodings: TeX, pMML, png See also: Annotations for §4.14 and Ch.4
4.14.5		$\csc z$	$={\displaystyle \frac{1}{\sin z}},$
ⓘ Defines: $\csc z$: cosecant function Symbols: $\sin z$: sine function and $z$: complex variable A&S Ref: 4.3.4 Permalink: http://dlmf.nist.gov/4.14.E5 Encodings: TeX, pMML, png See also: Annotations for §4.14 and Ch.4
4.14.6		$\sec z$	$={\displaystyle \frac{1}{\cos z}},$
ⓘ Defines: $\sec z$: secant function Symbols: $\cos z$: cosine function and $z$: complex variable A&S Ref: 4.3.5 Permalink: http://dlmf.nist.gov/4.14.E6 Encodings: TeX, pMML, png See also: Annotations for §4.14 and Ch.4
4.14.7		$\cot z$	$={\displaystyle \frac{\cos z}{\sin z}}={\displaystyle \frac{1}{\tan z}}.$
ⓘ Defines: $\cot z$: cotangent function Symbols: $\cos z$: cosine function, $\sin z$: sine function, $\tan z$: tangent function and $z$: complex variable A&S Ref: 4.3.6 Referenced by: §4.14, §4.45(i), §4.45(ii) Permalink: http://dlmf.nist.gov/4.14.E7 Encodings: TeX, pMML, png See also: Annotations for §4.14 and Ch.4

The functions $\sin z$ and $\cos z$ are entire. In $ℂ$ the zeros of $\sin z$ are $z=k\pi$, $k\in \mathbb{Z}$; the zeros of $\cos z$ are $z=\left(k+\frac{1}{2}\right)\pi$, $k\in \mathbb{Z}$. The functions $\tan z$, $\csc z$, $\sec z$, and $\cot z$ are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7).

For $k\in \mathbb{Z}$

4.14.8		$\sin \left(z+2k\pi \right)$	$=\sin z,$
ⓘ 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: TeX, pMML, png See also: Annotations for §4.14 and Ch.4
4.14.9		$\cos \left(z+2k\pi \right)$	$=\cos z,$
ⓘ 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: TeX, pMML, png See also: Annotations for §4.14 and Ch.4
4.14.10		$\tan \left(z+k\pi \right)$	$=\tan z.$
ⓘ 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: TeX, pMML, png See also: Annotations for §4.14 and Ch.4

4.13 Lambert $W$-Function4.15 Graphics

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