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.

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