Categories: Complex analysis, Mathematics, Quantum mechanics.

# Sokhotski-Plemelj theorem

The goal is to evaluate integrals of the following form, where $f(x)$ 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 \frac{x}{x^2 + \eta^2} f(x) \dd{x} - i \int_a^b \frac{\eta}{x^2 + \eta^2} f(x) \dd{x} \end{aligned}

In the real part, notice that the integrand diverges for $x \to 0$ when $\eta \to 0^+$; more specifically, there is a singularity at zero. We therefore 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 $\delta(x)$:

\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) Sokhotski-Plemelj theorem of complex analysis, which is important in quantum mechanics:

\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} \frac{1}{x} - i \pi \delta(x) \end{aligned}

That was the real version of the theorem, which is a special case of a general result in complex analysis. Consider the following function:

\begin{aligned} \phi(z) = \oint_C \frac{f(\zeta)}{\zeta - z} \dd{\zeta} \end{aligned}

Where $f(z)$ must be holomorphic. For all $z$ not on $C$, this $\phi(z)$ exists, but not for $z \in C$, since the integral diverges then. However, in the limit when approaching $C$, we can still obtain a value for $\phi$, with a caveat: the value depends on the direction we approach $C$ from! The full Sokhotski-Plemelj theorem then states, for all $z$ on the closed contour $C$:

\begin{aligned} \boxed{ \lim_{y \to z} \phi(y) = \mathcal{P} \oint_C \frac{f(\zeta)}{\zeta - z} \dd{\zeta} \pm \: i \pi f(z) } \end{aligned}

Where $\pm$ is $+$ if $C$ is approached from the inside, and $-$ if from outside. The above real version follows by making $C$ an infinitely large semicircle with its flat side on the real line: the integrand vanishes away from the real axis, because $1 / (\zeta \!-\! z) \to 0$ for $|\zeta| \to \infty$.