addition formula

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

Examples

1. Addition formula of an additive function $f$,
   $f(x+y)=f(x)+f(y)$

2. Addition formula of the natural power function, i.e. the binomial theorem,
   ${(x+y)}^n={\sum }_{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right)x^{n-j}{y}^j\mathit{\hspace{1em}\hspace{1em}}(n=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots })$

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

4. Addition formulae of the trigonometric functions (http://planetmath.org/DefinitionsInTrigonometry), e.g.
   $\cos (x+y)=\cos x\cos y-\sin x\sin y,\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x\tan y}$

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

6. Addition formula of the Bessel function,
   $J_n(x+y)={\sum }_{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{\nu }(x)J_{n-\nu }(y)\mathit{\hspace{1em}\hspace{1em}}(n=0,\pm 1,\pm 2,\mathrm{\dots })$

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 formula
Canonical name	AdditionFormula
Date of creation	2013-03-22 14:44:20
Last modified on	2013-03-22 14:44:20
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	26
Author	pahio (2872)
Entry type	Definition
Classification	msc 30D05
Classification	msc 30A99
Classification	msc 26A99
Synonym	addition theorem
Defines	addition formulae
Defines	subtraction formula
Defines	subtraction formulae