Physics Town

continues

Posted on October 6, 2007 by rpflee

Generator of Translation

$e^{a \frac{d}{dx}} = \sum_{n=0}^\infty \frac{1}{n!} a^n \left( \frac{d}{dx}\right)^n f(x)$

$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$

$f(x+a) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} x^n$

$f(a+x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(x)}{n!} a^n$

$= \sum_{n=0}^{\infty} \frac{1}{n!} a^n \left( \frac{d}{dx} \right)^n f(x)$

$e^{a \frac{d}{dx}} f(x) = f(x+a)$

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2007.10.06 Saturday (c) CHK2