4 Elementary Functions Trigonometric Functions 4.13 Lambert

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§4.14 Definitions and Periodicity

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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	$\displaystyle\sin z$	$\displaystyle=\frac{e^{\mathrm{i}z}-e^{-\mathrm{i}z}}{2\mathrm{i}},$
ⓘ Defines: $\sin\NVar{z}$ : 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	$\displaystyle\cos z$	$\displaystyle=\frac{e^{\mathrm{i}z}+e^{-\mathrm{i}z}}{2},$
ⓘ Defines: $\cos\NVar{z}$ : 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	$\displaystyle\cos z\pm i\sin z$	$\displaystyle=e^{\pm iz},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : 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	$\displaystyle\tan z$	$\displaystyle=\frac{\sin z}{\cos z},$
ⓘ Defines: $\tan\NVar{z}$ : tangent function Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : 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	$\displaystyle\csc z$	$\displaystyle=\frac{1}{\sin z},$
ⓘ Defines: $\csc\NVar{z}$ : cosecant function Symbols: $\sin\NVar{z}$ : 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	$\displaystyle\sec z$	$\displaystyle=\frac{1}{\cos z},$
ⓘ Defines: $\sec\NVar{z}$ : secant function Symbols: $\cos\NVar{z}$ : 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	$\displaystyle\cot z$	$\displaystyle=\frac{\cos z}{\sin z}=\frac{1}{\tan z}.$
ⓘ Defines: $\cot\NVar{z}$ : cotangent function Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : 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 $\mathbb{C}$ the zeros of $\sin z$ are $z=k\pi$ , $k\in\mathbb{Z}$ ; the zeros of $\cos z$ are $z=\left(k+\tfrac{1}{2}\right)\pi$ , $k\in\mathbb{Z}$ . 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	$\displaystyle\sin\left(z+2k\pi\right)$	$\displaystyle=\sin z,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : 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	$\displaystyle\cos\left(z+2k\pi\right)$	$\displaystyle=\cos z,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : 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	$\displaystyle\tan\left(z+k\pi\right)$	$\displaystyle=\tan z.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\tan\NVar{z}$ : 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

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