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Homework Help: Problem with functional

  1. Apr 5, 2010 #1
    I have the following relation:

    [tex] W_{\varepsilon}[J] = \mathrm{exp} \left[ - \varepsilon \int \mathrm{d} x \, \left( \dfrac{\delta}{\delta J(x)} \right)^{n} \right] \mathrm{exp}(W[J]) [/tex]

    where W is a functional of J.

    Now I read in a textbook that it follows

    [tex] W_{\varepsilon}[J] = W[J] - \varepsilon \mathrm{e}^{-W[J]} \int \mathrm{d} x \, \left( \dfrac{\delta}{\delta J(x)} \right)^{n} \mathrm{e}^{W[J]} + \mathcal{O}(\varepsilon^{2}) [/tex].

    Unfortunately I absolutely don't know how to obtain this result. Maybe some kind of Baker-Campbell-Hausdorff formula is used here or a relation such as
    [tex] \mathrm{e}^{A} B \mathrm{e}^{-A} = B + [A,B] + ... [/tex]
    but nevertheless I don't know how to obtain the result above. Does anyone have an idea?
  2. jcsd
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