4 Elementary Functions Hyperbolic Functions 4.27 Sums 4.29 Graphics

§4.28 Definitions and Periodicity

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Keywords:: definitions, hyperbolic functions
Notes:: See Hobson (1928, pp. 322–326), Levinson and Redheffer (1970, pp. 56–57).
Permalink:: http://dlmf.nist.gov/4.28
See also:: Annotations for Ch.4

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

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Keywords:: hyperbolic functions, relations to hyperbolic functions, relations to trigonometric functions, trigonometric functions
See also:: Annotations for §4.28 and Ch.4

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

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Keywords:: analytic properties, hyperbolic functions, periodicity, poles, trigonometric functions, zeros
See also:: Annotations for §4.28 and Ch.4

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, $k\in\mathbb{Z}$ .

4.27 Sums 4.29 Graphics

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