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

§4.35 Identities

Keywords:: hyperbolic functions
Referenced by:: ¶ ‣ §3.10(ii)
Permalink:: http://dlmf.nist.gov/4.35

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

§4.35(i) Addition Formulas

Notes:: See Hobson (1928, pp. 323–324).
Keywords:: hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.35.i

4.35.1 $\mathop{\sinh\/}\nolimits\!\left(u\pm v\right)=\mathop{\sinh\/}\nolimits u% \mathop{\cosh\/}\nolimits v\pm\mathop{\cosh\/}\nolimits u\mathop{\sinh\/}% \nolimits v,$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function
A&S Ref:: 4.5.24 (modified)
Permalink:: http://dlmf.nist.gov/4.35.E1
Encodings:: TeX, pMML, png

4.35.2 $\mathop{\cosh\/}\nolimits\!\left(u\pm v\right)=\mathop{\cosh\/}\nolimits u% \mathop{\cosh\/}\nolimits v\pm\mathop{\sinh\/}\nolimits u\mathop{\sinh\/}% \nolimits v,$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function
A&S Ref:: 4.5.25 (modified)
Permalink:: http://dlmf.nist.gov/4.35.E2
Encodings:: TeX, pMML, png

4.35.3 $\mathop{\tanh\/}\nolimits\!\left(u\pm v\right)=\frac{\mathop{\tanh\/}\nolimits u% \pm\mathop{\tanh\/}\nolimits v}{1\pm\mathop{\tanh\/}\nolimits u\mathop{\tanh\/% }\nolimits v},$

Symbols:: $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function
A&S Ref:: 4.5.26 (modified)
Permalink:: http://dlmf.nist.gov/4.35.E3
Encodings:: TeX, pMML, png

4.35.4 $\mathop{\coth\/}\nolimits\!\left(u\pm v\right)=\frac{\pm\mathop{\coth\/}% \nolimits u\mathop{\coth\/}\nolimits v+1}{\mathop{\coth\/}\nolimits u\pm% \mathop{\coth\/}\nolimits v}.$

Symbols:: $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function
A&S Ref:: 4.5.27 (modified)
Permalink:: http://dlmf.nist.gov/4.35.E4
Encodings:: TeX, pMML, png

4.35.5 $\mathop{\sinh\/}\nolimits u+\mathop{\sinh\/}\nolimits v=2\mathop{\sinh\/}% \nolimits\!\left(\frac{u+v}{2}\right)\mathop{\cosh\/}\nolimits\!\left(\frac{u-% v}{2}\right),$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function
A&S Ref:: 4.5.41
Permalink:: http://dlmf.nist.gov/4.35.E5
Encodings:: TeX, pMML, png

4.35.6 $\mathop{\sinh\/}\nolimits u-\mathop{\sinh\/}\nolimits v=2\mathop{\cosh\/}% \nolimits\!\left(\frac{u+v}{2}\right)\mathop{\sinh\/}\nolimits\!\left(\frac{u-% v}{2}\right),$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function
A&S Ref:: 4.5.42
Permalink:: http://dlmf.nist.gov/4.35.E6
Encodings:: TeX, pMML, png

4.35.7 $\mathop{\cosh\/}\nolimits u+\mathop{\cosh\/}\nolimits v=2\mathop{\cosh\/}% \nolimits\!\left(\frac{u+v}{2}\right)\mathop{\cosh\/}\nolimits\!\left(\frac{u-% v}{2}\right),$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function
A&S Ref:: 4.5.43
Permalink:: http://dlmf.nist.gov/4.35.E7
Encodings:: TeX, pMML, png

4.35.8 $\mathop{\cosh\/}\nolimits u-\mathop{\cosh\/}\nolimits v=2\mathop{\sinh\/}% \nolimits\!\left(\frac{u+v}{2}\right)\mathop{\sinh\/}\nolimits\!\left(\frac{u-% v}{2}\right),$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function
A&S Ref:: 4.5.44
Permalink:: http://dlmf.nist.gov/4.35.E8
Encodings:: TeX, pMML, png

4.35.9 $\mathop{\tanh\/}\nolimits u\pm\mathop{\tanh\/}\nolimits v=\frac{\mathop{\sinh% \/}\nolimits\!\left(u\pm v\right)}{\mathop{\cosh\/}\nolimits u\mathop{\cosh\/}% \nolimits v},$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function and $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function
A&S Ref:: 4.5.45
Permalink:: http://dlmf.nist.gov/4.35.E9
Encodings:: TeX, pMML, png

4.35.10 $\mathop{\coth\/}\nolimits u\pm\mathop{\coth\/}\nolimits v=\frac{\mathop{\sinh% \/}\nolimits\!\left(v\pm u\right)}{\mathop{\sinh\/}\nolimits u\mathop{\sinh\/}% \nolimits v}.$

Symbols:: $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function and $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function
A&S Ref:: 4.5.46 (modified)
Permalink:: http://dlmf.nist.gov/4.35.E10
Encodings:: TeX, pMML, png

§4.35(ii) Squares and Products

Notes:: See Hobson (1928, p. 323).
Keywords:: hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.35.ii

4.35.11 ${\mathop{\cosh\/}\nolimits^{{2}}}z-{\mathop{\sinh\/}\nolimits^{{2}}}z=1,$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function and : complex variable
A&S Ref:: 4.5.16
Permalink:: http://dlmf.nist.gov/4.35.E11
Encodings:: TeX, pMML, png

4.35.12 ${\mathop{\mathrm{sech}\/}\nolimits^{{2}}}z=1-{\mathop{\tanh\/}\nolimits^{{2}}}z,$

Symbols:: $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function and : complex variable
A&S Ref:: 4.5.17
Permalink:: http://dlmf.nist.gov/4.35.E12
Encodings:: TeX, pMML, png

4.35.13 ${\mathop{\mathrm{csch}\/}\nolimits^{{2}}}z={\mathop{\coth\/}\nolimits^{{2}}}z-1.$

Symbols:: $\mathop{\mathrm{csch}\/}\nolimits z$ : hyperbolic cosecant function, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function and : complex variable
A&S Ref:: 4.5.18
Permalink:: http://dlmf.nist.gov/4.35.E13
Encodings:: TeX, pMML, png

4.35.14 $2\mathop{\sinh\/}\nolimits u\mathop{\sinh\/}\nolimits v=\mathop{\cosh\/}% \nolimits\!\left(u+v\right)-\mathop{\cosh\/}\nolimits\!\left(u-v\right),$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function
A&S Ref:: 4.5.38
Permalink:: http://dlmf.nist.gov/4.35.E14
Encodings:: TeX, pMML, png

4.35.15 $2\mathop{\cosh\/}\nolimits u\mathop{\cosh\/}\nolimits v=\mathop{\cosh\/}% \nolimits\!\left(u+v\right)+\mathop{\cosh\/}\nolimits\!\left(u-v\right),$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function
A&S Ref:: 4.5.39
Permalink:: http://dlmf.nist.gov/4.35.E15
Encodings:: TeX, pMML, png

4.35.16 $2\mathop{\sinh\/}\nolimits u\mathop{\cosh\/}\nolimits v=\mathop{\sinh\/}% \nolimits\!\left(u+v\right)+\mathop{\sinh\/}\nolimits\!\left(u-v\right).$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function
A&S Ref:: 4.5.40
Permalink:: http://dlmf.nist.gov/4.35.E16
Encodings:: TeX, pMML, png

4.35.17 ${\mathop{\sinh\/}\nolimits^{{2}}}u-{\mathop{\sinh\/}\nolimits^{{2}}}v=\mathop{% \sinh\/}\nolimits\!\left(u+v\right)\mathop{\sinh\/}\nolimits\!\left(u-v\right),$

Symbols:: $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function
A&S Ref:: 4.5.47
Permalink:: http://dlmf.nist.gov/4.35.E17
Encodings:: TeX, pMML, png

4.35.18 ${\mathop{\cosh\/}\nolimits^{{2}}}u-{\mathop{\cosh\/}\nolimits^{{2}}}v=\mathop{% \sinh\/}\nolimits\!\left(u+v\right)\mathop{\sinh\/}\nolimits\!\left(u-v\right),$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function
A&S Ref:: 4.5.47
Permalink:: http://dlmf.nist.gov/4.35.E18
Encodings:: TeX, pMML, png

4.35.19 ${\mathop{\sinh\/}\nolimits^{{2}}}u+{\mathop{\cosh\/}\nolimits^{{2}}}v=\mathop{% \cosh\/}\nolimits\!\left(u+v\right)\mathop{\cosh\/}\nolimits\!\left(u-v\right).$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function
A&S Ref:: 4.5.48
Permalink:: http://dlmf.nist.gov/4.35.E19
Encodings:: TeX, pMML, png

§4.35(iii) Multiples of the Argument

Notes:: See Hobson (1928, pp. 324–325).
Keywords:: hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.35.iii

4.35.20 $\mathop{\sinh\/}\nolimits\frac{z}{2}=\left(\frac{\mathop{\cosh\/}\nolimits z-1% }{2}\right)^{{1/2}},$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function and : complex variable
A&S Ref:: 4.5.28
Permalink:: http://dlmf.nist.gov/4.35.E20
Encodings:: TeX, pMML, png

4.35.21 $\mathop{\cosh\/}\nolimits\frac{z}{2}=\left(\frac{\mathop{\cosh\/}\nolimits z+1% }{2}\right)^{{1/2}},$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and : complex variable
A&S Ref:: 4.5.29
Permalink:: http://dlmf.nist.gov/4.35.E21
Encodings:: TeX, pMML, png

4.35.22 $\mathop{\tanh\/}\nolimits\frac{z}{2}=\left(\frac{\mathop{\cosh\/}\nolimits z-1% }{\mathop{\cosh\/}\nolimits z+1}\right)^{{1/2}}=\frac{\mathop{\cosh\/}% \nolimits z-1}{\mathop{\sinh\/}\nolimits z}=\frac{\mathop{\sinh\/}\nolimits z}% {\mathop{\cosh\/}\nolimits z+1}.$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function and : complex variable
A&S Ref:: 4.5.30
Permalink:: http://dlmf.nist.gov/4.35.E22
Encodings:: TeX, pMML, png

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

4.35.23 $\mathop{\sinh\/}\nolimits\!\left(-z\right)=-\mathop{\sinh\/}\nolimits z,$

Symbols:: $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function and : complex variable
A&S Ref:: 4.5.21
Permalink:: http://dlmf.nist.gov/4.35.E23
Encodings:: TeX, pMML, png

4.35.24 $\mathop{\cosh\/}\nolimits\!\left(-z\right)=\mathop{\cosh\/}\nolimits z,$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and : complex variable
A&S Ref:: 4.5.22
Permalink:: http://dlmf.nist.gov/4.35.E24
Encodings:: TeX, pMML, png

4.35.25 $\mathop{\tanh\/}\nolimits\!\left(-z\right)=-\mathop{\tanh\/}\nolimits z.$

Symbols:: $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function and : complex variable
A&S Ref:: 4.5.23
Permalink:: http://dlmf.nist.gov/4.35.E25
Encodings:: TeX, pMML, png

4.35.26 $\mathop{\sinh\/}\nolimits\!\left(2z\right)=2\mathop{\sinh\/}\nolimits z\mathop% {\cosh\/}\nolimits z=\frac{2\mathop{\tanh\/}\nolimits z}{1-{\mathop{\tanh\/}% \nolimits^{{2}}}z},$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function and : complex variable
A&S Ref:: 4.5.31
Permalink:: http://dlmf.nist.gov/4.35.E26
Encodings:: TeX, pMML, png

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:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function and : complex variable
A&S Ref:: 4.5.32
Permalink:: http://dlmf.nist.gov/4.35.E27
Encodings:: TeX, pMML, png

4.35.28 $\mathop{\tanh\/}\nolimits\!\left(2z\right)=\frac{2\mathop{\tanh\/}\nolimits z}% {1+{\mathop{\tanh\/}\nolimits^{{2}}}z},$

Symbols:: $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function and : complex variable
A&S Ref:: 4.5.33
Permalink:: http://dlmf.nist.gov/4.35.E28
Encodings:: TeX, pMML, png

4.35.29 $\mathop{\sinh\/}\nolimits\!\left(3z\right)=3\mathop{\sinh\/}\nolimits z+4{% \mathop{\sinh\/}\nolimits^{{3}}}z,$

Symbols:: $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function and : complex variable
A&S Ref:: 4.5.34
Permalink:: http://dlmf.nist.gov/4.35.E29
Encodings:: TeX, pMML, png

4.35.30 $\mathop{\cosh\/}\nolimits\!\left(3z\right)=-3\mathop{\cosh\/}\nolimits z+4{% \mathop{\cosh\/}\nolimits^{{3}}}z,$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and : complex variable
A&S Ref:: 4.5.35
Permalink:: http://dlmf.nist.gov/4.35.E30
Encodings:: TeX, pMML, png

4.35.31 $\mathop{\sinh\/}\nolimits\!\left(4z\right)=4{\mathop{\sinh\/}\nolimits^{{3}}}z% \mathop{\cosh\/}\nolimits z+4{\mathop{\cosh\/}\nolimits^{{3}}}z\mathop{\sinh\/% }\nolimits z,$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function and : complex variable
A&S Ref:: 4.5.36
Permalink:: http://dlmf.nist.gov/4.35.E31
Encodings:: TeX, pMML, png

4.35.32 $\mathop{\cosh\/}\nolimits\!\left(4z\right)={\mathop{\cosh\/}\nolimits^{{4}}}z+% 6{\mathop{\sinh\/}\nolimits^{{2}}}z{\mathop{\cosh\/}\nolimits^{{2}}}z+{\mathop% {\sinh\/}\nolimits^{{4}}}z.$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function and : complex variable
A&S Ref:: 4.5.37
Permalink:: http://dlmf.nist.gov/4.35.E32
Encodings:: TeX, pMML, png

4.35.33 $\mathop{\cosh\/}\nolimits\!\left(nz\right)\pm\mathop{\sinh\/}\nolimits\!\left(% nz\right)=(\mathop{\cosh\/}\nolimits z\pm\mathop{\sinh\/}\nolimits z)^{n},$ $n\in\Integer$ .

Symbols:: $\in$ : element of, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\Integer$ : set of all integers, : integer and : complex variable
A&S Ref:: 4.5.53
Permalink:: http://dlmf.nist.gov/4.35.E33
Encodings:: TeX, pMML, png

§4.35(iv) Real and Imaginary Parts; Moduli

Notes:: See Hobson (1928, p. 331).
Keywords:: hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.35.iv

With

4.35.34 $\mathop{\sinh\/}\nolimits z=\mathop{\sinh\/}\nolimits x\mathop{\cos\/}% \nolimits y+i\mathop{\cosh\/}\nolimits x\mathop{\sin\/}\nolimits y,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable, : real variable and : complex variable
A&S Ref:: 4.5.49
Permalink:: http://dlmf.nist.gov/4.35.E34
Encodings:: TeX, pMML, png

4.35.35 $\mathop{\cosh\/}\nolimits z=\mathop{\cosh\/}\nolimits x\mathop{\cos\/}% \nolimits y+i\mathop{\sinh\/}\nolimits x\mathop{\sin\/}\nolimits y,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable, : real variable and : complex variable
A&S Ref:: 4.5.50
Permalink:: http://dlmf.nist.gov/4.35.E35
Encodings:: TeX, pMML, png

4.35.36 $\mathop{\tanh\/}\nolimits z=\frac{\mathop{\sinh\/}\nolimits\!\left(2x\right)+i% \mathop{\sin\/}\nolimits\!\left(2y\right)}{\mathop{\cosh\/}\nolimits\!\left(2x% \right)+\mathop{\cos\/}\nolimits\!\left(2y\right)},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable, : real variable and : complex variable
A&S Ref:: 4.5.51
Permalink:: http://dlmf.nist.gov/4.35.E36
Encodings:: TeX, pMML, png

4.35.37 $\mathop{\coth\/}\nolimits z=\frac{\mathop{\sinh\/}\nolimits\!\left(2x\right)-i% \mathop{\sin\/}\nolimits\!\left(2y\right)}{\mathop{\cosh\/}\nolimits\!\left(2x% \right)-\mathop{\cos\/}\nolimits\!\left(2y\right)}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable, : real variable and : complex variable
A&S Ref:: 4.5.52
Permalink:: http://dlmf.nist.gov/4.35.E37
Encodings:: TeX, pMML, png

4.35.38 $|\mathop{\sinh\/}\nolimits z|=({\mathop{\sinh\/}\nolimits^{{2}}}x+{\mathop{% \sin\/}\nolimits^{{2}}}y)^{{1/2}}=\left(\tfrac{1}{2}(\mathop{\cosh\/}\nolimits% \!\left(2x\right)-\mathop{\cos\/}\nolimits\!\left(2y\right))\right)^{{1/2}},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable, : real variable and : complex variable
A&S Ref:: 4.5.54
Permalink:: http://dlmf.nist.gov/4.35.E38
Encodings:: TeX, pMML, png

4.35.39 $|\mathop{\cosh\/}\nolimits z|=({\mathop{\sinh\/}\nolimits^{{2}}}x+{\mathop{% \cos\/}\nolimits^{{2}}}y)^{{1/2}}=\left(\tfrac{1}{2}(\mathop{\cosh\/}\nolimits% \!\left(2x\right)+\mathop{\cos\/}\nolimits\!\left(2y\right))\right)^{{1/2}},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, : real variable, : real variable and : complex variable
A&S Ref:: 4.5.56
Permalink:: http://dlmf.nist.gov/4.35.E39
Encodings:: TeX, pMML, png

4.35.40 $|\mathop{\tanh\/}\nolimits z|=\left(\frac{\mathop{\cosh\/}\nolimits\!\left(2x% \right)-\mathop{\cos\/}\nolimits\!\left(2y\right)}{\mathop{\cosh\/}\nolimits\!% \left(2x\right)+\mathop{\cos\/}\nolimits\!\left(2y\right)}\right)^{{1/2}}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function, : real variable, : real variable and : complex variable
A&S Ref:: 4.5.58
Permalink:: http://dlmf.nist.gov/4.35.E40
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 4.34 Derivatives and Differential Equations 4.36 Infinite Products and Partial Fractions

§4.35 Identities

Contents

§4.35(i) Addition Formulas

§4.35(ii) Squares and Products

§4.35(iii) Multiples of the Argument

§4.35(iv) Real and Imaginary Parts; Moduli