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.
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.
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.
6. 6.

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  $e^{x-y}=\frac{e^{x}}{e^{y}}$.