Order of Integration (calculus) - Principal-value Integrals

Principal-value Integrals

For application to principal-value integrals, see Whittaker and Watson, Gakhov, Lu, or Zwillinger. See also the discussion of the Poincaré-Bertrand transformation in Obolashvili. An example where the order of integration cannot be exchanged is given by Kanwal:

while:

The second form is evaluated using a partial fraction expansion and an evaluation using the Sokhotski–Plemelj formula:

The notation indicates a Cauchy principal value. See Kanwal.