# addition formulas

The of a real (http://planetmath.org/RealFunction) or complex function shows how the value of the function 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 function $f$,
$f(x\!+\!y)=f(x)+f(y)$

2. 2.

Addition formula of the natural power function, i.e. the binomial theorem,
$(x\!+\!y)^{n}=\sum_{\nu=0}^{n}{n\choose\nu}x^{\nu}y^{n-\nu}\qquad(n=0,\,1,\,2,% \,\ldots)$

3. 3.

Addition formula of the exponential function (http://planetmath.org/ComplexExponentialFunction),
$e^{x+y}=e^{x}e^{y}$

4. 4.

Addition formulae of the trigonometric functions (http://planetmath.org/DefinitionsInTrigonometry), e.g.
$\cos(x\!+\!y)=\cos{x}\cos{y}-\sin{x}\sin{y},\footnote{The addition formula of % cosine is sometimes called the mother of all formulae''.}\,\,\,\,\tan(x\!+\!% y)=\frac{\tan{x}+\tan{y}}{1-\tan{x}\tan{y}}$

5. 5.

Addition formulae of the hyperbolic functions, e.g.
$\sinh(x\!+\!y)=\sinh{x}\cosh{y}+\cosh{x}\sinh{y}$

6. 6.

Addition formula of the Bessel function,
$J_{n}(x\!+\!y)=\sum_{\nu=-\infty}^{\infty}J_{\nu}(x)J_{n-\nu}(y)\qquad(n=0,\,% \pm 1,\,\pm 2,\,\ldots)$

The five first of those are instances of ; e.g. $\cosh{x}$  and  $\sinh{x}$  are tied together by the algebraic connection (http://planetmath.org/UnitHyperbola)  $\cosh^{2}{x}-\sinh^{2}{x}=1$.

One may also speak of the subtraction formulae of functions — one example would be  $e^{x-y}=\frac{e^{x}}{e^{y}}$.

 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