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| Title of object: |
addition formula |
| Canonical Name: |
AdditionFormula |
| Type: |
Definition |
| Created on: |
2004-10-15 03:56:32 |
| Modified on: |
2007-01-14 04:24:07 |
| Classification: |
msc:26A99, msc:30A99, msc:30D05 |
| Keywords: |
algebraic addition formula |
| Defines: |
addition formulae, subtraction formula, subtraction formulae |
Revision comment (for changes between this and next version):
Preamble:
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Content:
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 a homogeneous function $f$ of \PMlinkescapetext{degree} 1,\\
$f(x+y) = f(x)+f(y)$
\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.\\
$\cos(x+y) = \cos{x}\cos{y}-\sin{x}\sin{y},\,\,\,\,
\tan(x+y) = \frac{\tan{x}+\tan{y}}{1-\tan{x}\tan{y}}$
\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\, $e^{x-y} = \frac{e^x}{e^y}$. |
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