# §4.14 Definitions and Periodicity

 4.14.1 $\displaystyle\mathop{\sin\/}\nolimits z$ $\displaystyle=\frac{e^{\mathrm{i}z}-e^{-\mathrm{i}z}}{2\mathrm{i}},$ Defines: $\mathop{\sin\/}\nolimits\NVar{z}$: sine function Symbols: $\mathrm{e}$: base of exponential function and $z$: complex variable A&S Ref: 4.3.1 Referenced by: §4.45(ii) Permalink: http://dlmf.nist.gov/4.14.E1 Encodings: TeX, pMML, png See also: Annotations for 4.14 4.14.2 $\displaystyle\mathop{\cos\/}\nolimits z$ $\displaystyle=\frac{e^{\mathrm{i}z}+e^{-\mathrm{i}z}}{2},$ Defines: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function Symbols: $\mathrm{e}$: base of exponential function and $z$: complex variable A&S Ref: 4.3.2 Permalink: http://dlmf.nist.gov/4.14.E2 Encodings: TeX, pMML, png See also: Annotations for 4.14 4.14.3 $\displaystyle\mathop{\cos\/}\nolimits z\pm i\mathop{\sin\/}\nolimits z$ $\displaystyle=e^{\pm iz},$ 4.14.4 $\displaystyle\mathop{\tan\/}\nolimits z$ $\displaystyle=\frac{\mathop{\sin\/}\nolimits z}{\mathop{\cos\/}\nolimits z},$ Defines: $\mathop{\tan\/}\nolimits\NVar{z}$: tangent function Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sin\/}\nolimits\NVar{z}$: 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 4.14.5 $\displaystyle\mathop{\csc\/}\nolimits z$ $\displaystyle=\frac{1}{\mathop{\sin\/}\nolimits z},$ Defines: $\mathop{\csc\/}\nolimits\NVar{z}$: cosecant function Symbols: $\mathop{\sin\/}\nolimits\NVar{z}$: 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 4.14.6 $\displaystyle\mathop{\sec\/}\nolimits z$ $\displaystyle=\frac{1}{\mathop{\cos\/}\nolimits z},$ Defines: $\mathop{\sec\/}\nolimits\NVar{z}$: secant function Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: 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 4.14.7 $\displaystyle\mathop{\cot\/}\nolimits z$ $\displaystyle=\frac{\mathop{\cos\/}\nolimits z}{\mathop{\sin\/}\nolimits z}=% \frac{1}{\mathop{\tan\/}\nolimits z}.$ Defines: $\mathop{\cot\/}\nolimits\NVar{z}$: cotangent function Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sin\/}\nolimits\NVar{z}$: sine function, $\mathop{\tan\/}\nolimits\NVar{z}$: 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

The functions $\mathop{\sin\/}\nolimits z$ and $\mathop{\cos\/}\nolimits z$ are entire. In $\mathbb{C}$ the zeros of $\mathop{\sin\/}\nolimits z$ are $z=k\pi$, $k\in\mathbb{Z}$; the zeros of $\mathop{\cos\/}\nolimits z$ are $z=\left(k+\tfrac{1}{2}\right)\pi$, $k\in\mathbb{Z}$. The functions $\mathop{\tan\/}\nolimits z$, $\mathop{\csc\/}\nolimits z$, $\mathop{\sec\/}\nolimits z$, and $\mathop{\cot\/}\nolimits z$ 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 $\displaystyle\mathop{\sin\/}\nolimits\!\left(z+2k\pi\right)$ $\displaystyle=\mathop{\sin\/}\nolimits z,$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\sin\/}\nolimits\NVar{z}$: 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 4.14.9 $\displaystyle\mathop{\cos\/}\nolimits\!\left(z+2k\pi\right)$ $\displaystyle=\mathop{\cos\/}\nolimits z,$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\cos\/}\nolimits\NVar{z}$: 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 4.14.10 $\displaystyle\mathop{\tan\/}\nolimits\!\left(z+k\pi\right)$ $\displaystyle=\mathop{\tan\/}\nolimits z.$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\tan\/}\nolimits\NVar{z}$: 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