GATE EC 2022 Set 1 Q22 — Complex Analysis

A simple closed path $C$ in the complex plane is shown in the figure. If

$$\oint_{c} \frac{2^{z}}{z^{2} – 1}dz = – i \pi A,$$

where $i = \sqrt{-1},$ then the value of $A$ is _________ (rounded off to two decimal places).