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.
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