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DLMF: §4.35 Identities ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions

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4 Elementary Functions Hyperbolic Functions 4.34 Derivatives and Differential Equations 4.36 Infinite Products and Partial Fractions

§4.35 Identities

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Keywords:: hyperbolic functions, identities
Referenced by:: §3.10(ii)
Permalink:: http://dlmf.nist.gov/4.35
See also:: Annotations for Ch.4

Contents

§4.35(i) Addition Formulas

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Keywords:: addition formulas, hyperbolic functions
Notes:: See Hobson (1928, pp. 323–324).
Permalink:: http://dlmf.nist.gov/4.35.i
See also:: Annotations for §4.35 and Ch.4

4.35.1		$\displaystyle\sinh\left(u\pm v\right)$	$\displaystyle=\sinh u\cosh v\pm\cosh u\sinh v,$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function and $\sinh\NVar{z}$ : hyperbolic sine function A&S Ref: 4.5.24 (modified) Permalink: http://dlmf.nist.gov/4.35.E1 Encodings: TeX, pMML, png See also: Annotations for §4.35(i), §4.35 and Ch.4
4.35.2		$\displaystyle\cosh\left(u\pm v\right)$	$\displaystyle=\cosh u\cosh v\pm\sinh u\sinh v,$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function and $\sinh\NVar{z}$ : hyperbolic sine function A&S Ref: 4.5.25 (modified) Permalink: http://dlmf.nist.gov/4.35.E2 Encodings: TeX, pMML, png See also: Annotations for §4.35(i), §4.35 and Ch.4
4.35.3		$\displaystyle\tanh\left(u\pm v\right)$	$\displaystyle=\frac{\tanh u\pm\tanh v}{1\pm\tanh u\tanh v},$
ⓘ Symbols: $\tanh\NVar{z}$ : hyperbolic tangent function A&S Ref: 4.5.26 (modified) Permalink: http://dlmf.nist.gov/4.35.E3 Encodings: TeX, pMML, png See also: Annotations for §4.35(i), §4.35 and Ch.4
4.35.4		$\displaystyle\coth\left(u\pm v\right)$	$\displaystyle=\frac{\pm\coth u\coth v+1}{\coth u\pm\coth v}.$
ⓘ Symbols: $\coth\NVar{z}$ : hyperbolic cotangent function A&S Ref: 4.5.27 (modified) Permalink: http://dlmf.nist.gov/4.35.E4 Encodings: TeX, pMML, png See also: Annotations for §4.35(i), §4.35 and Ch.4

4.35.5		$\displaystyle\sinh u+\sinh v$	$\displaystyle=2\sinh\left(\frac{u+v}{2}\right)\cosh\left(\frac{u-v}{2}\right),$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function and $\sinh\NVar{z}$ : hyperbolic sine function A&S Ref: 4.5.41 Permalink: http://dlmf.nist.gov/4.35.E5 Encodings: TeX, pMML, png See also: Annotations for §4.35(i), §4.35 and Ch.4
4.35.6		$\displaystyle\sinh u-\sinh v$	$\displaystyle=2\cosh\left(\frac{u+v}{2}\right)\sinh\left(\frac{u-v}{2}\right),$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function and $\sinh\NVar{z}$ : hyperbolic sine function A&S Ref: 4.5.42 Permalink: http://dlmf.nist.gov/4.35.E6 Encodings: TeX, pMML, png See also: Annotations for §4.35(i), §4.35 and Ch.4
4.35.7		$\displaystyle\cosh u+\cosh v$	$\displaystyle=2\cosh\left(\frac{u+v}{2}\right)\cosh\left(\frac{u-v}{2}\right),$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function A&S Ref: 4.5.43 Permalink: http://dlmf.nist.gov/4.35.E7 Encodings: TeX, pMML, png See also: Annotations for §4.35(i), §4.35 and Ch.4
4.35.8		$\displaystyle\cosh u-\cosh v$	$\displaystyle=2\sinh\left(\frac{u+v}{2}\right)\sinh\left(\frac{u-v}{2}\right),$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function and $\sinh\NVar{z}$ : hyperbolic sine function A&S Ref: 4.5.44 Permalink: http://dlmf.nist.gov/4.35.E8 Encodings: TeX, pMML, png See also: Annotations for §4.35(i), §4.35 and Ch.4
4.35.9		$\displaystyle\tanh u\pm\tanh v$	$\displaystyle=\frac{\sinh\left(u\pm v\right)}{\cosh u\cosh v},$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function and $\tanh\NVar{z}$ : hyperbolic tangent function A&S Ref: 4.5.45 Permalink: http://dlmf.nist.gov/4.35.E9 Encodings: TeX, pMML, png See also: Annotations for §4.35(i), §4.35 and Ch.4
4.35.10		$\displaystyle\coth u\pm\coth v$	$\displaystyle=\frac{\sinh\left(v\pm u\right)}{\sinh u\sinh v}.$
ⓘ Symbols: $\coth\NVar{z}$ : hyperbolic cotangent function and $\sinh\NVar{z}$ : hyperbolic sine function A&S Ref: 4.5.46 (modified) Permalink: http://dlmf.nist.gov/4.35.E10 Encodings: TeX, pMML, png See also: Annotations for §4.35(i), §4.35 and Ch.4

§4.35(ii) Squares and Products

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Keywords:: hyperbolic functions, squares and products
Notes:: See Hobson (1928, p. 323).
Permalink:: http://dlmf.nist.gov/4.35.ii
See also:: Annotations for §4.35 and Ch.4

4.35.11		${\cosh}^{2}z-{\sinh}^{2}z=1,$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function and $z$ : complex variable A&S Ref: 4.5.16 Permalink: http://dlmf.nist.gov/4.35.E11 Encodings: TeX, pMML, png See also: Annotations for §4.35(ii), §4.35 and Ch.4

4.35.12		${\operatorname{sech}}^{2}z=1-{\tanh}^{2}z,$
ⓘ Symbols: $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\tanh\NVar{z}$ : hyperbolic tangent function and $z$ : complex variable A&S Ref: 4.5.17 Permalink: http://dlmf.nist.gov/4.35.E12 Encodings: TeX, pMML, png See also: Annotations for §4.35(ii), §4.35 and Ch.4

4.35.13		${\operatorname{csch}}^{2}z={\coth}^{2}z-1.$
ⓘ Symbols: $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $\coth\NVar{z}$ : hyperbolic cotangent function and $z$ : complex variable A&S Ref: 4.5.18 Permalink: http://dlmf.nist.gov/4.35.E13 Encodings: TeX, pMML, png See also: Annotations for §4.35(ii), §4.35 and Ch.4

4.35.14		$\displaystyle 2\sinh u\sinh v$	$\displaystyle=\cosh\left(u+v\right)-\cosh\left(u-v\right),$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function and $\sinh\NVar{z}$ : hyperbolic sine function A&S Ref: 4.5.38 Permalink: http://dlmf.nist.gov/4.35.E14 Encodings: TeX, pMML, png See also: Annotations for §4.35(ii), §4.35 and Ch.4
4.35.15		$\displaystyle 2\cosh u\cosh v$	$\displaystyle=\cosh\left(u+v\right)+\cosh\left(u-v\right),$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function A&S Ref: 4.5.39 Permalink: http://dlmf.nist.gov/4.35.E15 Encodings: TeX, pMML, png See also: Annotations for §4.35(ii), §4.35 and Ch.4
4.35.16		$\displaystyle 2\sinh u\cosh v$	$\displaystyle=\sinh\left(u+v\right)+\sinh\left(u-v\right).$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function and $\sinh\NVar{z}$ : hyperbolic sine function A&S Ref: 4.5.40 Permalink: http://dlmf.nist.gov/4.35.E16 Encodings: TeX, pMML, png See also: Annotations for §4.35(ii), §4.35 and Ch.4

4.35.17		$\displaystyle{\sinh}^{2}u-{\sinh}^{2}v$	$\displaystyle=\sinh\left(u+v\right)\sinh\left(u-v\right),$
ⓘ Symbols: $\sinh\NVar{z}$ : hyperbolic sine function A&S Ref: 4.5.47 Permalink: http://dlmf.nist.gov/4.35.E17 Encodings: TeX, pMML, png See also: Annotations for §4.35(ii), §4.35 and Ch.4
4.35.18		$\displaystyle{\cosh}^{2}u-{\cosh}^{2}v$	$\displaystyle=\sinh\left(u+v\right)\sinh\left(u-v\right),$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function and $\sinh\NVar{z}$ : hyperbolic sine function A&S Ref: 4.5.47 Permalink: http://dlmf.nist.gov/4.35.E18 Encodings: TeX, pMML, png See also: Annotations for §4.35(ii), §4.35 and Ch.4
4.35.19		$\displaystyle{\sinh}^{2}u+{\cosh}^{2}v$	$\displaystyle=\cosh\left(u+v\right)\cosh\left(u-v\right).$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function and $\sinh\NVar{z}$ : hyperbolic sine function A&S Ref: 4.5.48 Permalink: http://dlmf.nist.gov/4.35.E19 Encodings: TeX, pMML, png See also: Annotations for §4.35(ii), §4.35 and Ch.4

§4.35(iii) Multiples of the Argument

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Keywords:: hyperbolic functions, multiples of argument
Notes:: See Hobson (1928, pp. 324–325).
Permalink:: http://dlmf.nist.gov/4.35.iii
See also:: Annotations for §4.35 and Ch.4

4.35.20		$\sinh\frac{z}{2}=\left(\frac{\cosh z-1}{2}\right)^{1/2},$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function and $z$ : complex variable A&S Ref: 4.5.28 Permalink: http://dlmf.nist.gov/4.35.E20 Encodings: TeX, pMML, png See also: Annotations for §4.35(iii), §4.35 and Ch.4

4.35.21		$\cosh\frac{z}{2}=\left(\frac{\cosh z+1}{2}\right)^{1/2},$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function and $z$ : complex variable A&S Ref: 4.5.29 Permalink: http://dlmf.nist.gov/4.35.E21 Encodings: TeX, pMML, png See also: Annotations for §4.35(iii), §4.35 and Ch.4

4.35.22		$\tanh\frac{z}{2}=\left(\frac{\cosh z-1}{\cosh z+1}\right)^{1/2}=\frac{\cosh z-% 1}{\sinh z}=\frac{\sinh z}{\cosh z+1}.$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\tanh\NVar{z}$ : hyperbolic tangent function and $z$ : complex variable A&S Ref: 4.5.30 Permalink: http://dlmf.nist.gov/4.35.E22 Encodings: TeX, pMML, png See also: Annotations for §4.35(iii), §4.35 and Ch.4

The square roots assume their principal value on the positive real axis, and are determined by continuity elsewhere.

4.35.23		$\displaystyle\sinh\left(-z\right)$	$\displaystyle=-\sinh z,$
ⓘ Symbols: $\sinh\NVar{z}$ : hyperbolic sine function and $z$ : complex variable A&S Ref: 4.5.21 Permalink: http://dlmf.nist.gov/4.35.E23 Encodings: TeX, pMML, png See also: Annotations for §4.35(iii), §4.35 and Ch.4
4.35.24		$\displaystyle\cosh\left(-z\right)$	$\displaystyle=\cosh z,$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function and $z$ : complex variable A&S Ref: 4.5.22 Permalink: http://dlmf.nist.gov/4.35.E24 Encodings: TeX, pMML, png See also: Annotations for §4.35(iii), §4.35 and Ch.4
4.35.25		$\displaystyle\tanh\left(-z\right)$	$\displaystyle=-\tanh z.$
ⓘ Symbols: $\tanh\NVar{z}$ : hyperbolic tangent function and $z$ : complex variable A&S Ref: 4.5.23 Permalink: http://dlmf.nist.gov/4.35.E25 Encodings: TeX, pMML, png See also: Annotations for §4.35(iii), §4.35 and Ch.4

4.35.26		$\sinh\left(2z\right)=2\sinh z\cosh z=\frac{2\tanh z}{1-{\tanh}^{2}z},$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\tanh\NVar{z}$ : hyperbolic tangent function and $z$ : complex variable A&S Ref: 4.5.31 Permalink: http://dlmf.nist.gov/4.35.E26 Encodings: TeX, pMML, png See also: Annotations for §4.35(iii), §4.35 and Ch.4

4.35.27		$\cosh\left(2z\right)=2{\cosh}^{2}z-1=2{\sinh}^{2}z+1\\ ={\cosh}^{2}z+{\sinh}^{2}z,$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function and $z$ : complex variable A&S Ref: 4.5.32 Permalink: http://dlmf.nist.gov/4.35.E27 Encodings: TeX, pMML, png See also: Annotations for §4.35(iii), §4.35 and Ch.4

4.35.28		$\tanh\left(2z\right)=\frac{2\tanh z}{1+{\tanh}^{2}z},$
ⓘ Symbols: $\tanh\NVar{z}$ : hyperbolic tangent function and $z$ : complex variable A&S Ref: 4.5.33 Permalink: http://dlmf.nist.gov/4.35.E28 Encodings: TeX, pMML, png See also: Annotations for §4.35(iii), §4.35 and Ch.4

4.35.29		$\sinh\left(3z\right)=3\sinh z+4{\sinh}^{3}z,$
ⓘ Symbols: $\sinh\NVar{z}$ : hyperbolic sine function and $z$ : complex variable A&S Ref: 4.5.34 Permalink: http://dlmf.nist.gov/4.35.E29 Encodings: TeX, pMML, png See also: Annotations for §4.35(iii), §4.35 and Ch.4

4.35.30		$\cosh\left(3z\right)=-3\cosh z+4{\cosh}^{3}z,$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function and $z$ : complex variable A&S Ref: 4.5.35 Permalink: http://dlmf.nist.gov/4.35.E30 Encodings: TeX, pMML, png See also: Annotations for §4.35(iii), §4.35 and Ch.4

4.35.31		$\displaystyle\sinh\left(4z\right)$	$\displaystyle=4{\sinh}^{3}z\cosh z+4{\cosh}^{3}z\sinh z,$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function and $z$ : complex variable A&S Ref: 4.5.36 Permalink: http://dlmf.nist.gov/4.35.E31 Encodings: TeX, pMML, png See also: Annotations for §4.35(iii), §4.35 and Ch.4
4.35.32		$\displaystyle\cosh\left(4z\right)$	$\displaystyle={\cosh}^{4}z+6{\sinh}^{2}z{\cosh}^{2}z+{\sinh}^{4}z.$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function and $z$ : complex variable A&S Ref: 4.5.37 Permalink: http://dlmf.nist.gov/4.35.E32 Encodings: TeX, pMML, png See also: Annotations for §4.35(iii), §4.35 and Ch.4

4.35.33		$\cosh\left(nz\right)\pm\sinh\left(nz\right)=(\cosh z\pm\sinh z)^{n},$
$n\in\mathbb{Z}$ .
ⓘ Symbols: $\in$ : element of, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathbb{Z}$ : set of all integers, $n$ : integer and $z$ : complex variable A&S Ref: 4.5.53 Permalink: http://dlmf.nist.gov/4.35.E33 Encodings: TeX, pMML, png See also: Annotations for §4.35(iii), §4.35 and Ch.4

§4.35(iv) Real and Imaginary Parts; Moduli

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Keywords:: hyperbolic functions, moduli, real and imaginary parts
Notes:: See Hobson (1928, p. 331).
Permalink:: http://dlmf.nist.gov/4.35.iv
See also:: Annotations for §4.35 and Ch.4

With $z=x+iy$

4.35.34		$\displaystyle\sinh z$	$\displaystyle=\sinh x\cos y+i\cosh x\sin y,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.5.49 Permalink: http://dlmf.nist.gov/4.35.E34 Encodings: TeX, pMML, png See also: Annotations for §4.35(iv), §4.35 and Ch.4
4.35.35		$\displaystyle\cosh z$	$\displaystyle=\cosh x\cos y+i\sinh x\sin y,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.5.50 Permalink: http://dlmf.nist.gov/4.35.E35 Encodings: TeX, pMML, png See also: Annotations for §4.35(iv), §4.35 and Ch.4
4.35.36		$\displaystyle\tanh z$	$\displaystyle=\frac{\sinh\left(2x\right)+i\sin\left(2y\right)}{\cosh\left(2x% \right)+\cos\left(2y\right)},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.5.51 Permalink: http://dlmf.nist.gov/4.35.E36 Encodings: TeX, pMML, png See also: Annotations for §4.35(iv), §4.35 and Ch.4
4.35.37		$\displaystyle\coth z$	$\displaystyle=\frac{\sinh\left(2x\right)-i\sin\left(2y\right)}{\cosh\left(2x% \right)-\cos\left(2y\right)}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\coth\NVar{z}$ : hyperbolic cotangent function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.5.52 Permalink: http://dlmf.nist.gov/4.35.E37 Encodings: TeX, pMML, png See also: Annotations for §4.35(iv), §4.35 and Ch.4

4.35.38		$\|\sinh z\|=({\sinh}^{2}x+{\sin}^{2}y)^{1/2}=\left(\tfrac{1}{2}(\cosh\left(2x% \right)-\cos\left(2y\right))\right)^{1/2},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\sin\NVar{z}$ : sine function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.5.54 Permalink: http://dlmf.nist.gov/4.35.E38 Encodings: TeX, pMML, png See also: Annotations for §4.35(iv), §4.35 and Ch.4

4.35.39		$\|\cosh z\|=({\sinh}^{2}x+{\cos}^{2}y)^{1/2}=\left(\tfrac{1}{2}(\cosh\left(2x% \right)+\cos\left(2y\right))\right)^{1/2},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.5.56 Permalink: http://dlmf.nist.gov/4.35.E39 Encodings: TeX, pMML, png See also: Annotations for §4.35(iv), §4.35 and Ch.4

4.35.40		$\|\tanh z\|=\left(\frac{\cosh\left(2x\right)-\cos\left(2y\right)}{\cosh\left(2x% \right)+\cos\left(2y\right)}\right)^{1/2}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.5.58 Permalink: http://dlmf.nist.gov/4.35.E40 Encodings: TeX, pMML, png See also: Annotations for §4.35(iv), §4.35 and Ch.4

4.34 Derivatives and Differential Equations 4.36 Infinite Products and Partial Fractions

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