addition formulas

The addition formula 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. 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,
   ${(x+y)}^n={\sum }_{\nu =0}^n\left(\genfrac{}{}{0pt}{}{n}{\nu }\right)x^{\nu }{y}^{n-\nu }\mathit{\hspace{1em}\hspace{1em}}(n=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots })$

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 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