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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., $f(x_1,x_2,g(x_1,x_2))$?

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.

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    $\begingroup$ Where is your functional defined? $\endgroup$ – Davide Giraudo Jan 27 '12 at 20:53
  • $\begingroup$ For simplicity consider R to R $\endgroup$ – Jorge Jan 27 '12 at 23:00
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    $\begingroup$ 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. $\endgroup$ – Fabian Jan 27 '12 at 23:09
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    $\begingroup$ I was hoping for a little more elaboration. $\endgroup$ – Jorge Jan 28 '12 at 2:30
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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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    $\begingroup$ What is the intuition in the "quadratic" term being $\frac{\delta^2 f}{\delta g(x') \delta g(x)}$ I would've expected to see: $\frac{\delta^2 f}{\delta^2 g(x) }$. Do you have a link of some kind on this? $\endgroup$ – frogeyedpeas Feb 2 '19 at 18:23
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I think this paper may be helpful to you.

"Taylor-series expansion of density functionals," by Matthias Ernzerhof. Phys. Rev. A 50, 4593 – Published 1 December 1994

Its abstract reads:

A Taylor-series expansion of the density functional F[n] is constructed from the given expansion for the energy E[v]. The formalism is used to investigate the noninteracting kinetic-energy functional for one-dimensional systems. Furthermore, a formal analysis of the Taylor expansion of the density functional for an interacting electron system at finite temperature is given and the relation between density-functional theory and the field-theoretical approach to the many-particle problem is enlightened.

You can also find a chain rule to work through the high order functional derivatives that you will need.

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