Let f be a continuous function in the region
\[ D = \{ z \ | | z | > R , 0 \leq \arg Z \leq \theta \} \quad \mathrm { w h e r e } \quad 0 \leq \theta \leq 2 \pi . \]
If there exists k such that \(\operatorname* { l i m } _ { z \to \infty } z f ( z ) = k\) for z in the region D. Show that
\[ \operatorname* { l i m } _ { R ^ { \prime } \longrightarrow \infty } \int _ { L } f ( z ) d z = i \theta k , \]
where L is the part of the circle \(| z | = R ^ { \prime }\) which lies in the region D.