# Help on method of iteration

1. Feb 21, 2012

Let $D(\epsilon)$ be an arbitrary but well-behaved (finite at $\epsilon_F$) density of states, and $f(\epsilon,\mu,T)$ be the Fermi distribution. Assume that $\mu\approx\epsilon_F$ at all temperatures of interest. $f(\epsilon,\mu,T=0)-f(\epsilon,\mu,T)$ is an odd function of $\epsilon-\mu$ and it is different from zero only about when $|\epsilon-\mu|\lesssim k_b T$. Use the method of iteration, the Taylor expansion $D(\epsilon)\approx D(\mu)+D'(\mu)(\epsilon-\mu)$ and the equation
$$\int_{\epsilon_F}^\mu D(\epsilon)\, \mathrm{d}\epsilon = \int_0^\infty D(\epsilon)[f(\epsilon,\mu,T=0)-f(\epsilon,\mu,T)]\, \mathrm{d}\epsilon$$
to find $\mu$ to the leading order in terms of $D(\epsilon_F)$, $D'(\epsilon_F)$, $k_B$, $T$, and numerical constants.
I'm really not very familiar with the method of iteration. I'm just looking for some hints on how to get started on this. I know that $\epsilon_f-\mu$ is going to be a small parameter. I'm not sure if I should use the linear expansion of $D(\epsilon)$ on both sides of the equation or just on one side. I have the feeling that I need to get $\epsilon_F-\mu$ as one side of the equation and do some kind of perturbation. I would love to hear any hints and tips or resources on the method of iteration. Where do I begin?