# §4.28 Definitions and Periodicity

 4.28.1 $\displaystyle\sinh z$ $\displaystyle=\frac{e^{z}-e^{-z}}{2},$ ⓘ Defines: $\sinh\NVar{z}$: hyperbolic sine function Symbols: $\mathrm{e}$: base of natural logarithm and $z$: complex variable A&S Ref: 4.5.1 Referenced by: §4.45(i), §4.45(ii) Permalink: http://dlmf.nist.gov/4.28.E1 Encodings: TeX, pMML, png See also: Annotations for §4.28 and Ch.4 4.28.2 $\displaystyle\cosh z$ $\displaystyle=\frac{e^{z}+e^{-z}}{2},$ ⓘ Defines: $\cosh\NVar{z}$: hyperbolic cosine function Symbols: $\mathrm{e}$: base of natural logarithm and $z$: complex variable A&S Ref: 4.5.2 Permalink: http://dlmf.nist.gov/4.28.E2 Encodings: TeX, pMML, png See also: Annotations for §4.28 and Ch.4 4.28.3 $\displaystyle\cosh z\pm\sinh z$ $\displaystyle=e^{\pm z},$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\cosh\NVar{z}$: hyperbolic cosine function, $\sinh\NVar{z}$: hyperbolic sine function and $z$: complex variable A&S Ref: 4.5.19 4.5.20 Permalink: http://dlmf.nist.gov/4.28.E3 Encodings: TeX, pMML, png See also: Annotations for §4.28 and Ch.4 4.28.4 $\displaystyle\tanh z$ $\displaystyle=\frac{\sinh z}{\cosh z},$ ⓘ Defines: $\tanh\NVar{z}$: hyperbolic tangent function Symbols: $\cosh\NVar{z}$: hyperbolic cosine function, $\sinh\NVar{z}$: hyperbolic sine function and $z$: complex variable A&S Ref: 4.5.3 Permalink: http://dlmf.nist.gov/4.28.E4 Encodings: TeX, pMML, png See also: Annotations for §4.28 and Ch.4 4.28.5 $\displaystyle\operatorname{csch}z$ $\displaystyle=\frac{1}{\sinh z},$ ⓘ Defines: $\operatorname{csch}\NVar{z}$: hyperbolic cosecant function Symbols: $\sinh\NVar{z}$: hyperbolic sine function and $z$: complex variable A&S Ref: 4.5.4 Permalink: http://dlmf.nist.gov/4.28.E5 Encodings: TeX, pMML, png See also: Annotations for §4.28 and Ch.4 4.28.6 $\displaystyle\operatorname{sech}z$ $\displaystyle=\frac{1}{\cosh z},$ ⓘ Defines: $\operatorname{sech}\NVar{z}$: hyperbolic secant function Symbols: $\cosh\NVar{z}$: hyperbolic cosine function and $z$: complex variable A&S Ref: 4.5.5 Permalink: http://dlmf.nist.gov/4.28.E6 Encodings: TeX, pMML, png See also: Annotations for §4.28 and Ch.4 4.28.7 $\displaystyle\coth z$ $\displaystyle=\frac{1}{\tanh z}.$ ⓘ Defines: $\coth\NVar{z}$: hyperbolic cotangent function Symbols: $\tanh\NVar{z}$: hyperbolic tangent function and $z$: complex variable A&S Ref: 4.5.6 Referenced by: §4.45(i), §4.45(ii) Permalink: http://dlmf.nist.gov/4.28.E7 Encodings: TeX, pMML, png See also: Annotations for §4.28 and Ch.4

## Relations to Trigonometric Functions

 4.28.8 $\displaystyle\sin\left(iz\right)$ $\displaystyle=i\sinh z,$ ⓘ Symbols: $\sinh\NVar{z}$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin\NVar{z}$: sine function and $z$: complex variable A&S Ref: 4.5.7 Referenced by: §22.10(ii), §4.29(ii), §4.29(ii), §4.33 Permalink: http://dlmf.nist.gov/4.28.E8 Encodings: TeX, pMML, png See also: Annotations for §4.28, §4.28 and Ch.4 4.28.9 $\displaystyle\cos\left(iz\right)$ $\displaystyle=\cosh z,$ ⓘ Symbols: $\cos\NVar{z}$: cosine function, $\cosh\NVar{z}$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit and $z$: complex variable A&S Ref: 4.5.8 Permalink: http://dlmf.nist.gov/4.28.E9 Encodings: TeX, pMML, png See also: Annotations for §4.28, §4.28 and Ch.4 4.28.10 $\displaystyle\tan\left(iz\right)$ $\displaystyle=i\tanh z,$ ⓘ Symbols: $\tanh\NVar{z}$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, $\tan\NVar{z}$: tangent function and $z$: complex variable A&S Ref: 4.5.9 Referenced by: §22.10(ii) Permalink: http://dlmf.nist.gov/4.28.E10 Encodings: TeX, pMML, png See also: Annotations for §4.28, §4.28 and Ch.4 4.28.11 $\displaystyle\csc\left(iz\right)$ $\displaystyle=-i\operatorname{csch}z,$ ⓘ Symbols: $\csc\NVar{z}$: cosecant function, $\operatorname{csch}\NVar{z}$: hyperbolic cosecant function, $\mathrm{i}$: imaginary unit and $z$: complex variable A&S Ref: 4.5.10 Permalink: http://dlmf.nist.gov/4.28.E11 Encodings: TeX, pMML, png See also: Annotations for §4.28, §4.28 and Ch.4 4.28.12 $\displaystyle\sec\left(iz\right)$ $\displaystyle=\operatorname{sech}z,$ ⓘ Symbols: $\operatorname{sech}\NVar{z}$: hyperbolic secant function, $\mathrm{i}$: imaginary unit, $\sec\NVar{z}$: secant function and $z$: complex variable A&S Ref: 4.5.11 Permalink: http://dlmf.nist.gov/4.28.E12 Encodings: TeX, pMML, png See also: Annotations for §4.28, §4.28 and Ch.4 4.28.13 $\displaystyle\cot\left(iz\right)$ $\displaystyle=-i\coth z.$ ⓘ Symbols: $\cot\NVar{z}$: cotangent function, $\coth\NVar{z}$: hyperbolic cotangent function, $\mathrm{i}$: imaginary unit and $z$: complex variable A&S Ref: 4.5.12 Referenced by: §4.29(ii), §4.33 Permalink: http://dlmf.nist.gov/4.28.E13 Encodings: TeX, pMML, png See also: Annotations for §4.28, §4.28 and Ch.4

As a consequence, many properties of the hyperbolic functions follow immediately from the corresponding properties of the trigonometric functions.

## Periodicity and Zeros

The functions $\sinh z$ and $\cosh z$ have period $2\pi i$, and $\tanh z$ has period $\pi i$. The zeros of $\sinh z$ and $\cosh z$ are $z=ik\pi$ and $z=i\left(k+\frac{1}{2}\right)\pi$, respectively, $k\in\mathbb{Z}$.