Assume f is continuous in the region: \(0 < | z - a | \leq R , \ 0 \leq \arg ( z - a ) \leq \beta _ { 0 } \ ( 0 < \beta _ { 0 } \leq 2 \pi )\)and the limit\(\operatorname* { l i m } _ { z \to a } ( z - a ) f ( z ) = A\) exists. Show that
\[ \operatorname * { l i m } _ { r 0 } \int _ { \gamma _ { r } } f ( z ) d z = i A \beta _ { 0 } \ , \]
where \(\gamma _ { r } : = \lbrace z \mid z = a + r e ^ { i t } , 0 \leq t \leq \beta _ { 0 } \rbrace\)