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I've searched the web for an answer to this, and haven't come up with anything definite. If someone could help, i' d be grateful. The question asks for the Taylor expansion of a functional. Thus, given a real functional $f(g(x))$, what is the Taylor expansion about a function $h(x)$; What if the function is multi-variate e.g. $f(x_1,x_2,g(x_1,x_2))$?

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Where is your functional defined? –  Davide Giraudo Jan 27 '12 at 20:53
    
For simplicity consider R to R –  Jorge Jan 27 '12 at 23:00
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Try to search for functional derivative. In essence, you have to write $f(h(x) + \epsilon \delta g(x))$ and then perform a normal Taylor expansion in $\epsilon$ setting $\epsilon=1$ at the end (but keeping $\delta g(x)$ small. –  Fabian Jan 27 '12 at 23:09
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I was hoping for a little more elaboration. –  Jorge Jan 28 '12 at 2:30
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2 Answers

This is a physicist answer and is thus not expressed in the better mathematical "vocab", but the Taylor expansion around $x_0$ of a functional $f$ of a function $g(x)$, i.e., of $f[g(x)]$ noted $f[g]$ for simplicity is:

$f[g]=f[g_0]+\int dx \frac{\delta f[g_0]}{\delta g(x)}\Delta g(x)+\frac{1}{2!}\int dx dx^\prime \frac{\delta^2f[g_0]}{\delta g(x^\prime)\delta g(x)}\Delta g(x^\prime)\Delta g(x) + ... $

with $g_0=g(x_0)$ and $\Delta g(x)=g(x)-g(x_0)$

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I think this paper may be helpful to you. You can also find a chain rule to work through the high order functional derivatives that you will need.

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I will check it out as soon as i have the time. Thanks! –  Jorge Mar 25 '12 at 0:11
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