Cauchy principal value
The Cauchy principal value , or just principal value, is a method for integrating problematic functions, i.e. functions with singularities, whose integrals would otherwise diverge.
Consider a function with a singularity at some finite , which is hampering attempts at integrating it. To resolve this, we define the Cauchy principal value as follows:
If instead has a singularity at postive infinity , then we define as follows:
And analogously for . If has singularities both at and at , then we simply combine the two previous cases, such that is given by:
And so on, until all problematic singularities have been dealt with.
In some situations, for example involving the Sokhotski-Plemelj theorem, the symbol is written without an integral, in which case the calculations are implicitly integrated.