Categories: Mathematics.

# Cauchy principal value

The Cauchy principal value $\mathcal{P}$, or just principal value, is a method for integrating problematic functions, i.e. functions with singularities, whose integrals would otherwise diverge.

Consider a function $f(x)$ with a singularity at some finite $x = b$, which is hampering attempts at integrating it. To resolve this, we define the Cauchy principal value $\mathcal{P}$ as follows:

\begin{aligned} \boxed{ \mathcal{P} \int_a^c f(x) \dd{x} = \lim_{\varepsilon \to 0^{+}} \!\bigg( \int_a^{b - \varepsilon} f(x) \dd{x} + \int_{b + \varepsilon}^c f(x) \dd{x} \bigg) } \end{aligned}

If $f(x)$ instead has a singularity at postive infinity $+\infty$, then we define $\mathcal{P}$ as follows:

\begin{aligned} \boxed{ \mathcal{P} \int_{a}^\infty f(x) \dd{x} = \lim_{c \to \infty} \!\bigg( \int_{a}^c f(x) \dd{x} \bigg) } \end{aligned}

And analogously for $-\infty$. If $f(x)$ has singularities both at $+\infty$ and at $b$, then we simply combine the two previous cases, such that $\mathcal{P}$ is given by:

\begin{aligned} \mathcal{P} \int_{a}^\infty f(x) \:dx = \lim_{c \to \infty} \lim_{\varepsilon \to 0^{+}} \!\bigg( \int_{a}^{b - \varepsilon} f(x) \:dx + \int_{b + \varepsilon}^{c} f(x) \:dx \bigg) \end{aligned}

And so on, until all problematic singularities have been dealt with.

In some situations, for example involving the Sokhotski-Plemelj theorem, the symbol $\mathcal{P}$ is written without an integral, in which case the calculations are implicitly integrated.