§4.14 Definitions and Periodicity

 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 $\displaystyle\cos z\pm i\sin z$ $\displaystyle=e^{\pm iz},$ 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