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.