Categories: Mathematics.

Cauchy principal value

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

P∫acf(x)dx=lim⁡ε→0+ ⁣(∫ab−εf(x)dx+∫b+εcf(x)dx)\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)f(x) instead has a singularity at postive infinity +∞+\infty, then we define P\mathcal{P} as follows:

P∫a∞f(x)dx=lim⁡c→∞ ⁣(∫acf(x)dx)\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)f(x) has singularities both at +∞+\infty and at bb, then we simply combine the two previous cases, such that P\mathcal{P} is given by:

P∫a∞f(x) dx=lim⁡c→∞lim⁡ε→0+ ⁣(∫ab−εf(x) dx+∫b+εcf(x) dx)\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 P\mathcal{P} is written without an integral, in which case the calculations are implicitly integrated.