Oct 122011

This week I had to brush up on Green’s functions and I invariably stumbled again on the math of the function

    \[\frac{1}{\epsilon + i s},\]

which appears over and over again when one works with the Green function

    \[\hat{G}(\epsilon) = \frac{1}{\epsilon + i s - \hat{H}}.\]

The first useful thing to know is the name of the often used identity

    \[\frac{1}{\epsilon + i s} = \frac{\rm P.V.}{\pi} \int_{-\infty}^\infty \frac{\Delta(\epsilon') d \epsilon'}{\epsilon - \epsilon'} - i \Delta(\epsilon),\]

where P.V. stands for principal value of the integral and the function \Delta(\epsilon) is defined as

    \[\Delta(\epsilon) = - {\rm Im} \frac{1}{\epsilon + i s}.\]

The integral form of this useful identity is known as the Sokhatsky–Weierstrass theorem. The integral appearing in the expression above is a Hilbert transform.

The second trick that I found useful deals with the computation of the residue for functions that can be written as a fraction. Say function f(z) = g(z) / h(z) has a pole in the complex plane at c. Then the residue of f(z) at c is simply given by

    \[{\rm Res}(f(z),c) = \frac{g(c)}{h'(c)}.\]

This result is useful to compute the residue of expressions like

    \[G(\epsilon) = \frac{1}{\epsilon - \Lambda(\epsilon)},\]

that often occur in Green’s functions

    \[{\rm Res}(G(\epsilon),\epsilon_{\rm pole}) = \frac{1}{1 - \Lambda'(\epsilon_{\rm pole})}.\]

 October 12, 2011  Uncategorized

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