# §4.28 Definitions and Periodicity

 4.28.1 $\displaystyle\mathop{\sinh\/}\nolimits z$ $\displaystyle=\frac{e^{z}-e^{-z}}{2},$ Defines: $\mathop{\sinh\/}\nolimits\NVar{z}$: hyperbolic sine function Symbols: $\mathrm{e}$: base of exponential function 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 4.28.2 $\displaystyle\mathop{\cosh\/}\nolimits z$ $\displaystyle=\frac{e^{z}+e^{-z}}{2},$ Defines: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function Symbols: $\mathrm{e}$: base of exponential function 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 4.28.3 $\displaystyle\mathop{\cosh\/}\nolimits z\pm\mathop{\sinh\/}\nolimits z$ $\displaystyle=e^{\pm z},$ Symbols: $\mathrm{e}$: base of exponential function, $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function, $\mathop{\sinh\/}\nolimits\NVar{z}$: 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 4.28.4 $\displaystyle\mathop{\tanh\/}\nolimits z$ $\displaystyle=\frac{\mathop{\sinh\/}\nolimits z}{\mathop{\cosh\/}\nolimits z},$ Defines: $\mathop{\tanh\/}\nolimits\NVar{z}$: hyperbolic tangent function Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function, $\mathop{\sinh\/}\nolimits\NVar{z}$: 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 4.28.5 $\displaystyle\mathop{\mathrm{csch}\/}\nolimits z$ $\displaystyle=\frac{1}{\mathop{\sinh\/}\nolimits z},$ Defines: $\mathop{\mathrm{csch}\/}\nolimits\NVar{z}$: hyperbolic cosecant function Symbols: $\mathop{\sinh\/}\nolimits\NVar{z}$: 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 4.28.6 $\displaystyle\mathop{\mathrm{sech}\/}\nolimits z$ $\displaystyle=\frac{1}{\mathop{\cosh\/}\nolimits z},$ Defines: $\mathop{\mathrm{sech}\/}\nolimits\NVar{z}$: hyperbolic secant function Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: 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 4.28.7 $\displaystyle\mathop{\coth\/}\nolimits z$ $\displaystyle=\frac{1}{\mathop{\tanh\/}\nolimits z}.$ Defines: $\mathop{\coth\/}\nolimits\NVar{z}$: hyperbolic cotangent function Symbols: $\mathop{\tanh\/}\nolimits\NVar{z}$: 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

## Relations to Trigonometric Functions

 4.28.8 $\displaystyle\mathop{\sin\/}\nolimits\!\left(iz\right)$ $\displaystyle=i\mathop{\sinh\/}\nolimits z,$ Symbols: $\mathop{\sinh\/}\nolimits\NVar{z}$: hyperbolic sine function, $\mathop{\sin\/}\nolimits\NVar{z}$: 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 4.28.9 $\displaystyle\mathop{\cos\/}\nolimits\!\left(iz\right)$ $\displaystyle=\mathop{\cosh\/}\nolimits z,$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function 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 4.28.10 $\displaystyle\mathop{\tan\/}\nolimits\!\left(iz\right)$ $\displaystyle=i\mathop{\tanh\/}\nolimits z,$ Symbols: $\mathop{\tanh\/}\nolimits\NVar{z}$: hyperbolic tangent function, $\mathop{\tan\/}\nolimits\NVar{z}$: 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 4.28.11 $\displaystyle\mathop{\csc\/}\nolimits\!\left(iz\right)$ $\displaystyle=-i\mathop{\mathrm{csch}\/}\nolimits z,$ Symbols: $\mathop{\csc\/}\nolimits\NVar{z}$: cosecant function, $\mathop{\mathrm{csch}\/}\nolimits\NVar{z}$: hyperbolic cosecant function 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 4.28.12 $\displaystyle\mathop{\sec\/}\nolimits\!\left(iz\right)$ $\displaystyle=\mathop{\mathrm{sech}\/}\nolimits z,$ Symbols: $\mathop{\mathrm{sech}\/}\nolimits\NVar{z}$: hyperbolic secant function, $\mathop{\sec\/}\nolimits\NVar{z}$: 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 4.28.13 $\displaystyle\mathop{\cot\/}\nolimits\!\left(iz\right)$ $\displaystyle=-i\mathop{\coth\/}\nolimits z.$ Symbols: $\mathop{\cot\/}\nolimits\NVar{z}$: cotangent function, $\mathop{\coth\/}\nolimits\NVar{z}$: hyperbolic cotangent function 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

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

## Periodicity and Zeros

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