Chapter 14.7, Problem 21P

In Example 4 we stated a rule for evaluating a contour integral when the contour passes through simple poles. We proved that the result was correct for $PV{\displaystyle {\int }_{\Gamma }\frac{e^{iz}}{z}}dz$ around the contour $\Gamma$ shown here.

(a) By following the same method (integrating around $C^{\prime }$ of Figure 7.3 and letting $r\to 0$ ) show that the result is correct if we replace $e^{ix}$ by any $f(z)$ which is analytic at $z=0.$

(b) Repeat the proof in (a) for $PV{\displaystyle {\int }_{\Gamma }\frac{f(z)}{(z-a)}}dz,\text{ a real }$ (that is, a pole on the $x$ axis), with f(z) analytic at $z=a.$