Categories: Complex analysis, Mathematics, Quantum mechanics.

Sokhotski-Plemelj theorem

The goal is to evaluate integrals of the following form, where \(a < 0 < b\), and \(f(x)\) is assumed to be continuous in the integration interval \([a, b]\):

\[\begin{aligned} \lim_{\eta \to 0^+} \int_a^b \frac{f(x)}{x + i \eta} \dd{x} \end{aligned}\]

To do so, we start by splitting the integrand into its real and imaginary parts (limit hidden):

\[\begin{aligned} \int_a^b \frac{f(x)}{x + i \eta} \dd{x} &= \int_a^b \frac{x - i \eta}{x^2 + \eta^2} f(x) \dd{x} = \int_a^b \bigg( \frac{x}{x^2 + \eta^2} - i \frac{\eta}{x^2 + \eta^2} \bigg) f(x) \dd{x} \end{aligned}\]

To evaluate the real part, we notice that for \(\eta \to 0^+\) the integrand diverges for \(x \to 0\), and thus split the integral as follows:

\[\begin{aligned} \lim_{\eta \to 0^+} \int_a^b \frac{x f(x)}{x^2 + \eta^2} \dd{x} &= \lim_{\eta \to 0^+} \bigg( \int_a^{-\eta} \frac{x f(x)}{x^2 + \eta^2} \dd{x} + \int_\eta^b \frac{x f(x)}{x^2 + \eta^2} \dd{x} \bigg) \end{aligned}\]

This is simply the definition of the Cauchy principal value \(\mathcal{P}\), so the real part is given by:

\[\begin{aligned} \lim_{\eta \to 0^+} \int_a^b \frac{x f(x)}{x^2 + \eta^2} \dd{x} &= \mathcal{P} \int_a^b \frac{x f(x)}{x^2} \dd{x} = \mathcal{P} \int_a^b \frac{f(x)}{x} \dd{x} \end{aligned}\]

Meanwhile, in the imaginary part, we substitute \(\eta\) for \(1 / m\), and introduce \(\pi\):

\[\begin{aligned} \lim_{\eta \to 0^+} \int_a^b \frac{\eta \: f(x)}{x^2 + \eta^2} \dd{x} &= \lim_{m \to +\infty} \frac{\pi}{\pi} \int_a^b \frac{1/m}{x^2 + 1/m^2} f(x) \dd{x} \\ &= \lim_{m \to +\infty} \frac{\pi}{\pi} \int_a^b \frac{m}{1 + m^2 x^2} f(x) \dd{x} \end{aligned}\]

The expression \(m / \pi (1 + m^2 x^2)\) is a so-called nascent delta function, meaning that in the limit \(m \to +\infty\) it converges to the Dirac delta function:

\[\begin{aligned} \lim_{\eta \to 0^+} \int_a^b \frac{\eta \: f(x)}{x^2 + \eta^2} \dd{x} &= \pi \int_a^b \delta(x) \: f(x) \dd{x} = \pi f(0) \end{aligned}\]

By combining the real and imaginary parts, we thus arrive at the (real version of the) so-called Sokhotski-Plemelj theorem of complex analysis:

\[\begin{aligned} \boxed{ \lim_{\eta \to 0^+} \int_a^b \frac{f(x)}{x + i \eta} \dd{x} = \mathcal{P} \int_a^b \frac{f(x)}{x} \dd{x} - i \pi f(0) } \end{aligned}\]

However, this theorem is often written in the following sloppy way, where \(\eta\) is defined up front to be small, the integral is hidden, and \(f(x)\) is set to \(1\). This awkwardly leaves \(\mathcal{P}\) behind:

\[\begin{aligned} \frac{1}{x + i \eta} = \mathcal{P} \Big( \frac{1}{x} \Big) - i \pi \delta(x) \end{aligned}\]

The full, complex version of the Sokhotski-Plemelj theorem evaluates integrals of the following form over a contour \(C\) in the complex plane:

\[\begin{aligned} \phi(z) = \frac{1}{2 \pi i} \oint_C \frac{f(\zeta)}{\zeta - z} \dd{\zeta} \end{aligned}\]

Where \(f(z)\) must be holomorphic. The Sokhotski-Plemelj theorem then states:

\[\begin{aligned} \boxed{ \lim_{w \to z} \phi(w) = \frac{1}{2 \pi i} \mathcal{P} \oint_C \frac{f(\zeta)}{\zeta - z} \dd{\zeta} \pm \frac{f(z)}{2} } \end{aligned}\]

Where the sign is positive if \(z\) is inside \(C\), and negative if it is outside. The real version follows by letting \(C\) follow the whole real axis, making \(C\) an infinitely large semicircle, so that the integrand vanishes away from the real axis, because \(1 / (\zeta \!-\! z) \to 0\) for \(|\zeta| \to \infty\).

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.