4 Elementary FunctionsHyperbolic Functions4.34 Derivatives and Differential Equations4.36 Infinite Products and Partial Fractions

§4.35 Identities

ⓘ

Keywords:

hyperbolic functions, identities

Referenced by:

§3.10(ii)

Permalink:

http://dlmf.nist.gov/4.35

See also:

Annotations for Ch.4

Contents

1. §4.35(i) Addition Formulas
2. §4.35(ii) Squares and Products
3. §4.35(iii) Multiples of the Argument
4. §4.35(iv) Real and Imaginary Parts; Moduli

§4.35(i) Addition Formulas

ⓘ

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		$\sinh \left(u\pm v\right)$	$=\sinh u\cosh v\pm \cosh u\sinh v,$
ⓘ Symbols: $\cosh z$: hyperbolic cosine function and $\sinh z$: 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		$\cosh \left(u\pm v\right)$	$=\cosh u\cosh v\pm \sinh u\sinh v,$
ⓘ Symbols: $\cosh z$: hyperbolic cosine function and $\sinh z$: 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		$\tanh \left(u\pm v\right)$	$={\displaystyle \frac{\tanh u\pm \tanh v}{1\pm \tanh u\tanh v}},$
ⓘ Symbols: $\tanh z$: 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		$\coth \left(u\pm v\right)$	$={\displaystyle \frac{\pm \coth u\coth v+1}{\coth u\pm \coth v}}.$
ⓘ Symbols: $\coth z$: 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		$\sinh u+\sinh v$	$=2\sinh \left({\displaystyle \frac{u+v}{2}}\right)\cosh \left({\displaystyle \frac{u-v}{2}}\right),$
ⓘ Symbols: $\cosh z$: hyperbolic cosine function and $\sinh z$: 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		$\sinh u-\sinh v$	$=2\cosh \left({\displaystyle \frac{u+v}{2}}\right)\sinh \left({\displaystyle \frac{u-v}{2}}\right),$
ⓘ Symbols: $\cosh z$: hyperbolic cosine function and $\sinh z$: 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		$\cosh u+\cosh v$	$=2\cosh \left({\displaystyle \frac{u+v}{2}}\right)\cosh \left({\displaystyle \frac{u-v}{2}}\right),$
ⓘ Symbols: $\cosh z$: 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		$\cosh u-\cosh v$	$=2\sinh \left({\displaystyle \frac{u+v}{2}}\right)\sinh \left({\displaystyle \frac{u-v}{2}}\right),$
ⓘ Symbols: $\cosh z$: hyperbolic cosine function and $\sinh z$: 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		$\tanh u\pm \tanh v$	$={\displaystyle \frac{\sinh \left(u\pm v\right)}{\cosh u\cosh v}},$
ⓘ Symbols: $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $\tanh z$: 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		$\coth u\pm \coth v$	$={\displaystyle \frac{\sinh \left(v\pm u\right)}{\sinh u\sinh v}}.$
ⓘ Symbols: $\coth z$: hyperbolic cotangent function and $\sinh z$: 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

ⓘ

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: $\cosh z$: hyperbolic cosine function, $\sinh z$: 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		$${\mathrm{sech}}^{2}z=1-{\tanh }^{2}z,$$
ⓘ Symbols: $\mathrm{sech}z$: hyperbolic secant function, $\tanh z$: 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		$${\mathrm{csch}}^{2}z={\coth }^{2}z-1.$$
ⓘ Symbols: $\mathrm{csch}z$: hyperbolic cosecant function, $\coth z$: 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		$2\sinh u\sinh v$	$=\cosh \left(u+v\right)-\cosh \left(u-v\right),$
ⓘ Symbols: $\cosh z$: hyperbolic cosine function and $\sinh z$: 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		$2\cosh u\cosh v$	$=\cosh \left(u+v\right)+\cosh \left(u-v\right),$
ⓘ Symbols: $\cosh z$: 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		$2\sinh u\cosh v$	$=\sinh \left(u+v\right)+\sinh \left(u-v\right).$
ⓘ Symbols: $\cosh z$: hyperbolic cosine function and $\sinh z$: 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		${\sinh }^{2}u-{\sinh }^{2}v$	$=\sinh \left(u+v\right)\sinh \left(u-v\right),$
ⓘ Symbols: $\sinh z$: 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		${\cosh }^{2}u-{\cosh }^{2}v$	$=\sinh \left(u+v\right)\sinh \left(u-v\right),$
ⓘ Symbols: $\cosh z$: hyperbolic cosine function and $\sinh z$: 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		${\sinh }^{2}u+{\cosh }^{2}v$	$=\cosh \left(u+v\right)\cosh \left(u-v\right).$
ⓘ Symbols: $\cosh z$: hyperbolic cosine function and $\sinh z$: 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

ⓘ

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: $\cosh z$: hyperbolic cosine function, $\sinh z$: 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: $\cosh z$: 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: $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\tanh z$: 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		$\sinh \left(-z\right)$	$=-\sinh z,$
ⓘ Symbols: $\sinh z$: 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		$\cosh \left(-z\right)$	$=\cosh z,$
ⓘ Symbols: $\cosh z$: 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		$\tanh \left(-z\right)$	$=-\tanh z.$
ⓘ Symbols: $\tanh z$: 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: $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\tanh z$: 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: $\cosh z$: hyperbolic cosine function, $\sinh z$: 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: $\tanh z$: 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: $\sinh z$: 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: $\cosh z$: 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		$\sinh \left(4z\right)$	$=4{\sinh }^{3}z\cosh z+4{\cosh }^{3}z\sinh z,$
ⓘ Symbols: $\cosh z$: hyperbolic cosine function, $\sinh z$: 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		$\cosh \left(4z\right)$	$={\cosh }^{4}z+6{\sinh }^{2}z{\cosh }^{2}z+{\sinh }^{4}z.$
ⓘ Symbols: $\cosh z$: hyperbolic cosine function, $\sinh z$: 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, $\cosh z$: hyperbolic cosine function, $\sinh z$: 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

ⓘ

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 $z=x+\mathrm{i}y$

4.35.34		$\sinh z$	$=\sinh x\cos y+\mathrm{i}\cosh x\sin y,$
ⓘ Symbols: $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: 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		$\cosh z$	$=\cosh x\cos y+\mathrm{i}\sinh x\sin y,$
ⓘ Symbols: $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: 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		$\tanh z$	$={\displaystyle \frac{\sinh \left(2x\right)+\mathrm{i}\sin \left(2y\right)}{\cosh \left(2x\right)+\cos \left(2y\right)}},$
ⓘ Symbols: $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, $\sin z$: 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		$\coth z$	$={\displaystyle \frac{\sinh \left(2x\right)-\mathrm{i}\sin \left(2y\right)}{\cosh \left(2x\right)-\cos \left(2y\right)}}.$
ⓘ Symbols: $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: 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		$$|\sinh z|={({\sinh }^{2}x+{\sin }^{2}y)}^{1/2}={\left(\frac{1}{2}(\cosh \left(2x\right)-\cos \left(2y\right))\right)}^{1/2},$$
ⓘ Symbols: $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\sin z$: 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		$$|\cosh z|={({\sinh }^{2}x+{\cos }^{2}y)}^{1/2}={\left(\frac{1}{2}(\cosh \left(2x\right)+\cos \left(2y\right))\right)}^{1/2},$$
ⓘ Symbols: $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: 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: $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\tanh z$: 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 Equations4.36 Infinite Products and Partial Fractions

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