Taylor expansion of functional
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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$\begingroup$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 $\Delta g(x)=g(x)-g(x_0)$
$\endgroup$ 3 $\begingroup$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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