The addition formula of a real 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,
   $e^{{x+y}}=e^{x}e^{y}$

4. 4.

   Addition formulae of the trigonometric functions, 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 algebraic addition formulae; e.g. $\cosh{x}$  and  $\sinh{x}$  are tied together by the algebraic connection  $\cosh^{2}{x}-\sinh^{2}{x}=1$.

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