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单词 HyperbolicIdentities
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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.

Titlehyperbolic identities
Canonical nameHyperbolicIdentities
Date of creation2013-03-22 17:50:42
Last modified on2013-03-22 17:50:42
OwnerWkbj79 (1863)
Last modified byWkbj79 (1863)
Numerical id11
AuthorWkbj79 (1863)
Entry typeTopic
Classificationmsc 33B10
Classificationmsc 26A09
Synonymhyperbolic formulas
Synonymhyperbolic formulae
Related topicGoniometricFormulae
Related topicAdditionAndSubtractionFormulasForHyperbolicFunctions
Related topicExamplesOfPeriodicFunctions
Related topicTaylorSeriesOfHyperbolicFunctions
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