addition formula AdditionFormula 2004-10-15 03:56:32 2009-04-04 22:21:23 Definition addition formulae subtraction formula subtraction formulae algebraic addition formula % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The {\em addition formula} of a \PMlinkname{real}{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. \textbf{Examples} \begin{enumerate} \item Addition formula of an additive function $f$,\\ $f(x\!+\!y) = f(x)+f(y)$ \item Addition formula of the natural power function, i.e. the binomial theorem,\\ $(x\!+\!y)^n = \sum_{j = 0}^n {n\choose j} x^{n-j}y^j\qquad(n = 0,\,1,\,2,\,\ldots)$ \item Addition formula of the \PMlinkname{exponential function}{ComplexExponentialFunction},\\ $e^{x+y} = e^xe^y$ \item Addition formulae of the \PMlinkname{trigonometric functions}{DefinitionsInTrigonometry}, e.g.\\ $\cos(x\!+\!y) = \cos{x}\cos{y}-\sin{x}\sin{y},\footnote{The addition formula of cosine is sometimes called ``the mother of all formulae''.}\,\,\,\, \tan(x\!+\!y) = \frac{\tan{x}+\tan{y}}{1-\tan{x}\tan{y}}$ \item Addition formulae of the hyperbolic functions, e.g.\\ $\sinh(x\!+\!y) = \sinh{x}\cosh{y}+\cosh{x}\sinh{y}$ \item Addition formula of the Bessel function,\\ $J_n(x\!+\!y) = \sum_{\nu=-\infty}^{\infty}J_\nu(x)J_{n-\nu}(y) \qquad(n = 0,\,\pm1,\,\pm2,\,\ldots)$ \end{enumerate} The five first of those are instances of \PMlinkescapetext{{\em algebraic addition formulae}}; e.g. $\cosh{x}$\, and \,$\sinh{x}$\, are tied together by the algebraic \PMlinkname{connection}{UnitHyperbola} \,$\cosh^2{x}-\sinh^2{x} = 1$. One may also speak of the {\em subtraction formulae} of functions --- one example would be\, $e^{x-y} = \frac{e^x}{e^y}$.