1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

8.5 • INDENTED CONTOUR INTEGRALS 3 17

In this section we show how to use residues to evaluate the Cauchy principal
value of the integral off over (-oo, oo) when the integrand f has simple poles
on the x-axis. We state our main results and then look at some examples before
giving proofs.

Remark 8.2 The formulas in these theorems give the Cauchy principal value
of the integral, which pays special attention to the manner in which any limits
are taken. They are similar to those in Sections 8.3 and 8.4, except here we add
one-half the value of each residue at the points t1, t2, ... , t1 on the x-axis. •

• EXAMPLE 8.20 Evaluate P.V. J~ 00 ,,itt.: 8 by us ing complex analysis.
  Solution The integrand
  z z
  f (z) = -z3 __ 8 = -(z--- 2 - )-(z+l+_i_v'3)= 3 - (z+l--iv'3)- 3 ~