How do I write taylor expansion as exponential function?

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1. May 1, 2015

dwellexity

How do I write taylor expansion of a function of x,y,z (not at origin) as an exponential function?
Please see the attached image. I need help with the cross terms. I don't know how to include them in the exponential function?

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2. May 1, 2015

stevendaryl

Staff Emeritus
Well, $f(x+y) = f(x) + y \frac{d}{dx} f + \frac{y^2}{2} \frac{d^2}{dx^2} f + ...$, which can be formally written as:

$f(x+y) = e^{y \frac{\partial}{\partial x}} f(x)$

or to be more physics-like,

$f(x+y) = e^{\frac{i}{\hbar} y p_x} f(x)$

This is discussed here:
http://en.wikipedia.org/wiki/Translation_operator_(quantum_mechanics)

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