hyperbolic identities


There are many formulas involving hyperbolic functionsDlmfMathworldPlanetmath, many of which are to formulas for trigonometric functionsDlmfMathworldPlanetmath. Below is a list of some of these formulas (usually for real argumentsMathworldPlanetmath).

  1. 1.

    Hyperbolic version of Pythagorean identities

    • cosh2x-sinh2x=1

    • 1-tanh2x=sech2x

    • coth2x-1=csch2x

  2. 2.

    Fractional identitiesPlanetmathPlanetmath

    • tanhx=sinhxcoshx

    • cothx=coshxsinhx

    • cothx=1tanhx

    • tanhx=1cothx

    • cschx=1sinhx

    • sechx=1coshx

  3. 3.

    Hyperbolic functions of a purely imaginary numberMathworldPlanetmath

    • sinh(ix)=isinx

    • cosh(ix)=cosx

    • tanh(ix)=itanx

    • cosh(ix)=icotx

    • csch(ix)=icscx

    • sech(ix)=secx

  4. 4.

    Addition formulasPlanetmathPlanetmath and subtraction formulas (http://planetmath.org/AdditionAndSubtractionFormulasForHyperbolicFunctions)

    • sinh(x±y)=sinhxcoshy±coshxsinhy

    • cosh(x±y)=coshxcoshy±sinhxsinhy

    • tanh(x±y)=tanhx±tanhy1±tanhxtanhy

  5. 5.

    Formulas for hyperbolic functions of a complex numberMathworldPlanetmathPlanetmath

    • sinh(x+iy)=sinhxcosy+icoshxsiny

    • cosh(x+iy)=coshxcosy+isinhxsiny

    • tanh(x+iy)=tanhx+itany1+itanhxtany

  6. 6.

    Opposite formulas

    • sinh(-x)=-sinhx

    • cosh(-x)=coshx

    • tanh(-x)=-tanhx

  7. 7.

    Double argument formulas

    • sinh(2x)=2sinhxcoshx

    • cosh(2x)=cosh2x+sinh2x=2cosh2x-1=1+2sinh2x

    • tanh(2x)=2tanhx1+tanh2x

  8. 8.

    Periodicity (http://planetmath.org/Periodic) formulas

    • sinh(z+2πi)=sinhz

    • cosh(z+2πi)=coshz

    • tanh(z+πi)=tanhz

    Cf (http://planetmath.org/Cf). the periodicity of exponential functionDlmfDlmfMathworldPlanetmathPlanetmath.

  9. 9.

    Exponential formulas (http://planetmath.org/ExponentialFunction)

    • coshx=ex+e-x2

    • sinhx=ex-e-x2

    • tanhx=ex-e-xex+e-x

    • ex=coshx+sinhx

    • e-x=coshx-sinhx

    Note that the first three formulas given in this are definitions.

Title hyperbolic identities
Canonical name HyperbolicIdentities
Date of creation 2013-03-22 17:50:42
Last modified on 2013-03-22 17:50:42
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 11
Author Wkbj79 (1863)
Entry type Topic
Classification msc 33B10
Classification msc 26A09
Synonym hyperbolic formulas
Synonym hyperbolic formulae
Related topic GoniometricFormulae
Related topic AdditionAndSubtractionFormulasForHyperbolicFunctions
Related topic ExamplesOfPeriodicFunctions
Related topic TaylorSeriesOfHyperbolicFunctions