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单词 AdditionAndSubtractionFormulasForHyperbolicFunctions
释义

addition and subtraction formulas for hyperbolic functions


The addition formulasPlanetmathPlanetmath for hyperbolic sineMathworldPlanetmath, hyperbolic cosine, and hyperbolic tangent will be achieved via brute .

sinh(x+y)=ex+y-e-(x+y)2
=exey-exe-y+exe-y-e-xe-y2
=ex(ey-e-y2)+e-y(ex-e-x2)
=(coshx+sinhx)sinhy+(coshy-sinhy)sinhx
=coshxsinhy+sinhxsinhy+sinhxcoshy-sinhxsinhy
=sinhxcoshy+coshxsinhy
cosh(x+y)=ex+y+e-(x+y)2
=exey-exe-y+exe-y+e-xe-y2
=ex(ey-e-y2)+e-y(ex+e-x2)
=(coshx+sinhx)sinhy+(coshy-sinhy)coshx
=coshxsinhy+sinhxsinhy+coshxcoshy-coshxsinhy
=coshxcoshy+sinhxsinhy
tanh(x+y)=sinh(x+y)cosh(x+y)
=sinhxcoshy+coshxsinhycoshxcoshy+sinhxsinhy
=sinhxcoshxcoshycoshy+coshxcoshxsinhycoshycoshxcoshxcoshycoshy+sinhxcoshxsinhycoshy
=tanhx+tanhy1+tanhxtanhy

Note that sinh and tanh are odd functionsMathworldPlanetmath and cosh is an even function, i.e. (http://planetmath.org/Ie) sinh(-t)=-sinht, tanh(-t)=-tanht, and cosh(-t)=cosht. These facts enable us to obtain the subtraction formulas.

sinh(x-y)=sinh(x+(-y))=sinhxcosh(-y)+coshxsinh(-y)=sinhxcoshy-coshxsinhy
cosh(x-y)=cosh(x+(-y))=coshxcosh(-y)+sinhxsinh(-y)=coshxcoshy-sinhxsinhy
tanh(x-y)=tanh(x+(-y))=tanhx+tanh(-y)1+tanhxtanh(-y)=tanhx-tanhy1-tanhxtanhy
Titleaddition and subtraction formulas for hyperbolic functions
Canonical nameAdditionAndSubtractionFormulasForHyperbolicFunctions
Date of creation2013-03-22 17:50:45
Last modified on2013-03-22 17:50:45
OwnerWkbj79 (1863)
Last modified byWkbj79 (1863)
Numerical id5
AuthorWkbj79 (1863)
Entry typeDerivation
Classificationmsc 26A09
Classificationmsc 33B10
Synonymaddition and subtraction formulae for hyperbolic functions
Synonymaddition formulas for hyperbolic functions
Synonymaddition formulae for hyperbolic functions
Synonymsubtraction formulas for hyperbolic functions
Synonymsubtraction formulae for hyperbolic functions
Synonymaddition form
Related topicAdditionFormula
Related topicHyperbolicIdentities
Related topicAdditionFormulas
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