Assume f is continuous in the region: \(x \ge x _ { 0 } , 0 \le y \le b\) and the limit
\[ \operatorname* { l i m } _ { x \to + \infty } f ( x + i y ) = A \]
exists uniformly with respect to y (independent of y). Show that
\[ \operatorname* { l i m } _ { x + \infty } \int _ { \gamma _ { x } } f ( z ) d z = i A b , \]
where \(\gamma _ { x } : = \{ z \mid z = x + i t , 0 \leq t \leq b \}\)