# Homework Help: Problem with functional

1. Apr 5, 2010

### parton

I have the following relation:

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

where W is a functional of J.

Now I read in a textbook that it follows

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

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
$$\mathrm{e}^{A} B \mathrm{e}^{-A} = B + [A,B] + ...$$
but nevertheless I don't know how to obtain the result above. Does anyone have an idea?