addition formulas


The addition formulaPlanetmathPlanetmath of a real (http://planetmath.org/RealFunction) or complex function shows how the value of the functionMathworldPlanetmath at a sum-formed variable can be expressed with the values of this function and perhaps of another function at the addends.

Examples

  1. 1.

    Addition formula of an additive functionMathworldPlanetmath f,
    f(x+y)=f(x)+f(y)

  2. 2.

    Addition formula of the natural power function, i.e. the binomial theoremMathworldPlanetmath,
    (x+y)n=ν=0n(nν)xνyn-ν  (n=0, 1, 2,)

  3. 3.

    Addition formula of the exponential functionDlmfDlmfMathworldPlanetmath (http://planetmath.org/ComplexExponentialFunction),
    ex+y=exey

  4. 4.

    Addition formulae of the trigonometric functionsDlmfMathworldPlanetmath (http://planetmath.org/DefinitionsInTrigonometry), e.g.
    cos(x+y)=cosxcosy-sinxsiny,11The addition formula of cosine is sometimes called “the mother of all formulae”.tan(x+y)=tanx+tany1-tanxtany

  5. 5.

    Addition formulae of the hyperbolic functionsDlmfMathworldPlanetmath, e.g.
    sinh(x+y)=sinhxcoshy+coshxsinhy

  6. 6.

    Addition formula of the Bessel functionDlmfMathworldPlanetmathPlanetmath,
    Jn(x+y)=ν=-Jν(x)Jn-ν(y)  (n=0,±1,±2,)

The five first of those are instances of ; e.g. coshx  and  sinhx  are tied together by the algebraic connection (http://planetmath.org/UnitHyperbola)  cosh2x-sinh2x=1.

One may also speak of the subtraction formulae of functions — one example would be  ex-y=exey.

Title addition formulas
Canonical name AdditionFormulas
Date of creation 2013-03-22 19:35:28
Last modified on 2013-03-22 19:35:28
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Definition
Classification msc 30D05
Classification msc 30A99
Classification msc 26A99
Related topic ExampleOnSolvingAFunctionalEquation
Related topic ProofOfAdditionFormulaOfExp
Related topic AdditionFormulasForSineAndCosine
Related topic AdditionFormulaForTangent
Related topic AdditionAndSubtractionFormulasForHyperbolicFunctions
Defines addition formula
Defines subtraction formula