# §4.35 Identities

###### Contents

 4.35.1 $\displaystyle\mathop{\sinh\/}\nolimits\!\left(u\pm v\right)$ $\displaystyle=\mathop{\sinh\/}\nolimits u\mathop{\cosh\/}\nolimits v\pm\mathop% {\cosh\/}\nolimits u\mathop{\sinh\/}\nolimits v,$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function and $\mathop{\sinh\/}\nolimits\NVar{z}$: 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) 4.35.2 $\displaystyle\mathop{\cosh\/}\nolimits\!\left(u\pm v\right)$ $\displaystyle=\mathop{\cosh\/}\nolimits u\mathop{\cosh\/}\nolimits v\pm\mathop% {\sinh\/}\nolimits u\mathop{\sinh\/}\nolimits v,$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function and $\mathop{\sinh\/}\nolimits\NVar{z}$: 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) 4.35.3 $\displaystyle\mathop{\tanh\/}\nolimits\!\left(u\pm v\right)$ $\displaystyle=\frac{\mathop{\tanh\/}\nolimits u\pm\mathop{\tanh\/}\nolimits v}% {1\pm\mathop{\tanh\/}\nolimits u\mathop{\tanh\/}\nolimits v},$ Symbols: $\mathop{\tanh\/}\nolimits\NVar{z}$: 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) 4.35.4 $\displaystyle\mathop{\coth\/}\nolimits\!\left(u\pm v\right)$ $\displaystyle=\frac{\pm\mathop{\coth\/}\nolimits u\mathop{\coth\/}\nolimits v+% 1}{\mathop{\coth\/}\nolimits u\pm\mathop{\coth\/}\nolimits v}.$ Symbols: $\mathop{\coth\/}\nolimits\NVar{z}$: 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)
 4.35.5 $\displaystyle\mathop{\sinh\/}\nolimits u+\mathop{\sinh\/}\nolimits v$ $\displaystyle=2\mathop{\sinh\/}\nolimits\!\left(\frac{u+v}{2}\right)\mathop{% \cosh\/}\nolimits\!\left(\frac{u-v}{2}\right),$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function and $\mathop{\sinh\/}\nolimits\NVar{z}$: 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) 4.35.6 $\displaystyle\mathop{\sinh\/}\nolimits u-\mathop{\sinh\/}\nolimits v$ $\displaystyle=2\mathop{\cosh\/}\nolimits\!\left(\frac{u+v}{2}\right)\mathop{% \sinh\/}\nolimits\!\left(\frac{u-v}{2}\right),$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function and $\mathop{\sinh\/}\nolimits\NVar{z}$: 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) 4.35.7 $\displaystyle\mathop{\cosh\/}\nolimits u+\mathop{\cosh\/}\nolimits v$ $\displaystyle=2\mathop{\cosh\/}\nolimits\!\left(\frac{u+v}{2}\right)\mathop{% \cosh\/}\nolimits\!\left(\frac{u-v}{2}\right),$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: 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) 4.35.8 $\displaystyle\mathop{\cosh\/}\nolimits u-\mathop{\cosh\/}\nolimits v$ $\displaystyle=2\mathop{\sinh\/}\nolimits\!\left(\frac{u+v}{2}\right)\mathop{% \sinh\/}\nolimits\!\left(\frac{u-v}{2}\right),$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function and $\mathop{\sinh\/}\nolimits\NVar{z}$: 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) 4.35.9 $\displaystyle\mathop{\tanh\/}\nolimits u\pm\mathop{\tanh\/}\nolimits v$ $\displaystyle=\frac{\mathop{\sinh\/}\nolimits\!\left(u\pm v\right)}{\mathop{% \cosh\/}\nolimits u\mathop{\cosh\/}\nolimits v},$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function, $\mathop{\sinh\/}\nolimits\NVar{z}$: hyperbolic sine function and $\mathop{\tanh\/}\nolimits\NVar{z}$: 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) 4.35.10 $\displaystyle\mathop{\coth\/}\nolimits u\pm\mathop{\coth\/}\nolimits v$ $\displaystyle=\frac{\mathop{\sinh\/}\nolimits\!\left(v\pm u\right)}{\mathop{% \sinh\/}\nolimits u\mathop{\sinh\/}\nolimits v}.$ Symbols: $\mathop{\coth\/}\nolimits\NVar{z}$: hyperbolic cotangent function and $\mathop{\sinh\/}\nolimits\NVar{z}$: 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(ii) Squares and Products

 4.35.11 ${\mathop{\cosh\/}\nolimits^{2}}z-{\mathop{\sinh\/}\nolimits^{2}}z=1,$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function, $\mathop{\sinh\/}\nolimits\NVar{z}$: hyperbolic sine function and $z$: complex variable A&S Ref: 4.5.16 Permalink: http://dlmf.nist.gov/4.35.E11 Encodings: TeX, pMML, png See also: Annotations for 4.35(ii)
 4.35.12 ${\mathop{\mathrm{sech}\/}\nolimits^{2}}z=1-{\mathop{\tanh\/}\nolimits^{2}}z,$ Symbols: $\mathop{\mathrm{sech}\/}\nolimits\NVar{z}$: hyperbolic secant function, $\mathop{\tanh\/}\nolimits\NVar{z}$: 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)
 4.35.13 ${\mathop{\mathrm{csch}\/}\nolimits^{2}}z={\mathop{\coth\/}\nolimits^{2}}z-1.$ Symbols: $\mathop{\mathrm{csch}\/}\nolimits\NVar{z}$: hyperbolic cosecant function, $\mathop{\coth\/}\nolimits\NVar{z}$: 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)
 4.35.14 $\displaystyle 2\mathop{\sinh\/}\nolimits u\mathop{\sinh\/}\nolimits v$ $\displaystyle=\mathop{\cosh\/}\nolimits\!\left(u+v\right)-\mathop{\cosh\/}% \nolimits\!\left(u-v\right),$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function and $\mathop{\sinh\/}\nolimits\NVar{z}$: 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) 4.35.15 $\displaystyle 2\mathop{\cosh\/}\nolimits u\mathop{\cosh\/}\nolimits v$ $\displaystyle=\mathop{\cosh\/}\nolimits\!\left(u+v\right)+\mathop{\cosh\/}% \nolimits\!\left(u-v\right),$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: 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) 4.35.16 $\displaystyle 2\mathop{\sinh\/}\nolimits u\mathop{\cosh\/}\nolimits v$ $\displaystyle=\mathop{\sinh\/}\nolimits\!\left(u+v\right)+\mathop{\sinh\/}% \nolimits\!\left(u-v\right).$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function and $\mathop{\sinh\/}\nolimits\NVar{z}$: 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)
 4.35.17 $\displaystyle{\mathop{\sinh\/}\nolimits^{2}}u-{\mathop{\sinh\/}\nolimits^{2}}v$ $\displaystyle=\mathop{\sinh\/}\nolimits\!\left(u+v\right)\mathop{\sinh\/}% \nolimits\!\left(u-v\right),$ Symbols: $\mathop{\sinh\/}\nolimits\NVar{z}$: 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) 4.35.18 $\displaystyle{\mathop{\cosh\/}\nolimits^{2}}u-{\mathop{\cosh\/}\nolimits^{2}}v$ $\displaystyle=\mathop{\sinh\/}\nolimits\!\left(u+v\right)\mathop{\sinh\/}% \nolimits\!\left(u-v\right),$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function and $\mathop{\sinh\/}\nolimits\NVar{z}$: 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) 4.35.19 $\displaystyle{\mathop{\sinh\/}\nolimits^{2}}u+{\mathop{\cosh\/}\nolimits^{2}}v$ $\displaystyle=\mathop{\cosh\/}\nolimits\!\left(u+v\right)\mathop{\cosh\/}% \nolimits\!\left(u-v\right).$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function and $\mathop{\sinh\/}\nolimits\NVar{z}$: 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(iii) Multiples of the Argument

 4.35.20 $\mathop{\sinh\/}\nolimits\frac{z}{2}=\left(\frac{\mathop{\cosh\/}\nolimits z-1% }{2}\right)^{1/2},$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function, $\mathop{\sinh\/}\nolimits\NVar{z}$: hyperbolic sine function and $z$: complex variable A&S Ref: 4.5.28 Permalink: http://dlmf.nist.gov/4.35.E20 Encodings: TeX, pMML, png See also: Annotations for 4.35(iii)
 4.35.21 $\mathop{\cosh\/}\nolimits\frac{z}{2}=\left(\frac{\mathop{\cosh\/}\nolimits z+1% }{2}\right)^{1/2},$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: 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)
 4.35.22 $\mathop{\tanh\/}\nolimits\frac{z}{2}=\left(\frac{\mathop{\cosh\/}\nolimits z-1% }{\mathop{\cosh\/}\nolimits z+1}\right)^{1/2}=\frac{\mathop{\cosh\/}\nolimits z% -1}{\mathop{\sinh\/}\nolimits z}=\frac{\mathop{\sinh\/}\nolimits z}{\mathop{% \cosh\/}\nolimits z+1}.$

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

 4.35.23 $\displaystyle\mathop{\sinh\/}\nolimits\!\left(-z\right)$ $\displaystyle=-\mathop{\sinh\/}\nolimits z,$ Symbols: $\mathop{\sinh\/}\nolimits\NVar{z}$: 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) 4.35.24 $\displaystyle\mathop{\cosh\/}\nolimits\!\left(-z\right)$ $\displaystyle=\mathop{\cosh\/}\nolimits z,$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: 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) 4.35.25 $\displaystyle\mathop{\tanh\/}\nolimits\!\left(-z\right)$ $\displaystyle=-\mathop{\tanh\/}\nolimits z.$ Symbols: $\mathop{\tanh\/}\nolimits\NVar{z}$: 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)
 4.35.26 $\mathop{\sinh\/}\nolimits\!\left(2z\right)=2\mathop{\sinh\/}\nolimits z\mathop% {\cosh\/}\nolimits z=\frac{2\mathop{\tanh\/}\nolimits z}{1-{\mathop{\tanh\/}% \nolimits^{2}}z},$
 4.35.27 $\mathop{\cosh\/}\nolimits\!\left(2z\right)=2{\mathop{\cosh\/}\nolimits^{2}}z-1% =2{\mathop{\sinh\/}\nolimits^{2}}z+1\\ ={\mathop{\cosh\/}\nolimits^{2}}z+{\mathop{\sinh\/}\nolimits^{2}}z,$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function, $\mathop{\sinh\/}\nolimits\NVar{z}$: hyperbolic sine function and $z$: complex variable A&S Ref: 4.5.32 Permalink: http://dlmf.nist.gov/4.35.E27 Encodings: TeX, pMML, png See also: Annotations for 4.35(iii)
 4.35.28 $\mathop{\tanh\/}\nolimits\!\left(2z\right)=\frac{2\mathop{\tanh\/}\nolimits z}% {1+{\mathop{\tanh\/}\nolimits^{2}}z},$ Symbols: $\mathop{\tanh\/}\nolimits\NVar{z}$: 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)
 4.35.29 $\mathop{\sinh\/}\nolimits\!\left(3z\right)=3\mathop{\sinh\/}\nolimits z+4{% \mathop{\sinh\/}\nolimits^{3}}z,$ Symbols: $\mathop{\sinh\/}\nolimits\NVar{z}$: 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)
 4.35.30 $\mathop{\cosh\/}\nolimits\!\left(3z\right)=-3\mathop{\cosh\/}\nolimits z+4{% \mathop{\cosh\/}\nolimits^{3}}z,$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: 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)
 4.35.31 $\displaystyle\mathop{\sinh\/}\nolimits\!\left(4z\right)$ $\displaystyle=4{\mathop{\sinh\/}\nolimits^{3}}z\mathop{\cosh\/}\nolimits z+4{% \mathop{\cosh\/}\nolimits^{3}}z\mathop{\sinh\/}\nolimits z,$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function, $\mathop{\sinh\/}\nolimits\NVar{z}$: hyperbolic sine function and $z$: complex variable A&S Ref: 4.5.36 Permalink: http://dlmf.nist.gov/4.35.E31 Encodings: TeX, pMML, png See also: Annotations for 4.35(iii) 4.35.32 $\displaystyle\mathop{\cosh\/}\nolimits\!\left(4z\right)$ $\displaystyle={\mathop{\cosh\/}\nolimits^{4}}z+6{\mathop{\sinh\/}\nolimits^{2}% }z{\mathop{\cosh\/}\nolimits^{2}}z+{\mathop{\sinh\/}\nolimits^{4}}z.$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function, $\mathop{\sinh\/}\nolimits\NVar{z}$: hyperbolic sine function and $z$: complex variable A&S Ref: 4.5.37 Permalink: http://dlmf.nist.gov/4.35.E32 Encodings: TeX, pMML, png See also: Annotations for 4.35(iii)
 4.35.33 $\mathop{\cosh\/}\nolimits\!\left(nz\right)\pm\mathop{\sinh\/}\nolimits\!\left(% nz\right)=(\mathop{\cosh\/}\nolimits z\pm\mathop{\sinh\/}\nolimits z)^{n},$ $n\in\mathbb{Z}$.

## §4.35(iv) Real and Imaginary Parts; Moduli

With $z=x+iy$

 4.35.34 $\displaystyle\mathop{\sinh\/}\nolimits z$ $\displaystyle=\mathop{\sinh\/}\nolimits x\mathop{\cos\/}\nolimits y+i\mathop{% \cosh\/}\nolimits x\mathop{\sin\/}\nolimits y,$ 4.35.35 $\displaystyle\mathop{\cosh\/}\nolimits z$ $\displaystyle=\mathop{\cosh\/}\nolimits x\mathop{\cos\/}\nolimits y+i\mathop{% \sinh\/}\nolimits x\mathop{\sin\/}\nolimits y,$ 4.35.36 $\displaystyle\mathop{\tanh\/}\nolimits z$ $\displaystyle=\frac{\mathop{\sinh\/}\nolimits\!\left(2x\right)+i\mathop{\sin\/% }\nolimits\!\left(2y\right)}{\mathop{\cosh\/}\nolimits\!\left(2x\right)+% \mathop{\cos\/}\nolimits\!\left(2y\right)},$ 4.35.37 $\displaystyle\mathop{\coth\/}\nolimits z$ $\displaystyle=\frac{\mathop{\sinh\/}\nolimits\!\left(2x\right)-i\mathop{\sin\/% }\nolimits\!\left(2y\right)}{\mathop{\cosh\/}\nolimits\!\left(2x\right)-% \mathop{\cos\/}\nolimits\!\left(2y\right)}.$
 4.35.38 $|\mathop{\sinh\/}\nolimits z|=({\mathop{\sinh\/}\nolimits^{2}}x+{\mathop{\sin% \/}\nolimits^{2}}y)^{1/2}=\left(\tfrac{1}{2}(\mathop{\cosh\/}\nolimits\!\left(% 2x\right)-\mathop{\cos\/}\nolimits\!\left(2y\right))\right)^{1/2},$
 4.35.39 $|\mathop{\cosh\/}\nolimits z|=({\mathop{\sinh\/}\nolimits^{2}}x+{\mathop{\cos% \/}\nolimits^{2}}y)^{1/2}=\left(\tfrac{1}{2}(\mathop{\cosh\/}\nolimits\!\left(% 2x\right)+\mathop{\cos\/}\nolimits\!\left(2y\right))\right)^{1/2},$
 4.35.40 $|\mathop{\tanh\/}\nolimits z|=\left(\frac{\mathop{\cosh\/}\nolimits\!\left(2x% \right)-\mathop{\cos\/}\nolimits\!\left(2y\right)}{\mathop{\cosh\/}\nolimits\!% \left(2x\right)+\mathop{\cos\/}\nolimits\!\left(2y\right)}\right)^{1/2}.$