Principal-value integrals - Revisited

The principal-value (PV) integral has proved a useful tool in many fields of physics. The PV is a specific method for obtaining a finite result for an improper integral. When the integration passes through a simple pole, one speaks of a "first-order" PV. In this paper, we examine first-order PV integrals and analyze several of their important properties. First, we discuss how the PV agrees with one's naive expectation about these integrals. Next, we show that the basic definition of the first-order PV gives a generalized formula for the complex-variable PV expression. Finally, we show the correspondence between the finite-limit PV integral of x(-1) along the real axis and the path integral of z(-1) (where z = x + iy) in the complex plane.