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I hope this question is not to vague. There is some relationship between the formal operator $$e^\frac{d}{dx} = 1 + \frac{d}{dx} + \frac{1}{2!}\frac{d^2}{dx^2} + \ldots $$ and taylor's formula. Is there a precise relationship that can be stated here? Specifically, I was taught a mnemonic once for taylors formula in multiple dimensions in terms of a similar exponential. Does anybody know what I am referring to? |
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In one dimension we have the Maclaurin series $$\begin{eqnarray*} f(x) &=& \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n \\ &=& \left.\exp\left(x \frac{d}{dx'}\right)f(x')\right|_{x'=0} \end{eqnarray*}$$ and, more generally, the Taylor series $$\begin{eqnarray*} f(x) &=& \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n \\ &=& \left.\exp\left[(x-a) \frac{d}{dx'}\right]f(x')\right|_{x'=a}. \end{eqnarray*}$$ Likewise, in $\mathbb{R}^n$ $$\begin{eqnarray*} f({\bf x}) &=& \left. \sum_{n=0}^\infty \frac{1}{n!} ({\bf x}\cdot \nabla_{{\bf x'}})^n f({\bf x'}) \right|_{{\bf x'}={\bf 0}} \\ &=& \left. \exp({\bf x}\cdot \nabla_{{\bf x'}}) f({\bf x'}) \right|_{{\bf x'}={\bf 0}} \end{eqnarray*}$$ and $$\begin{eqnarray*} f({\bf x}) &=& \left. \sum_{n=0}^\infty \frac{1}{n!} [({\bf x}-{\bf a})\cdot \nabla_{{\bf x'}}]^n f({\bf x'}) \right|_{{\bf x'}={\bf a}} \\ &=& \left. \exp[({\bf x}-{\bf a}) \cdot \nabla_{{\bf x'}}] f({\bf x'}) \right|_{{\bf x'}={\bf a}}. \end{eqnarray*}$$ |
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