GATE 2022 EC – Question 32
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).
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 0.49 to 0.51
Explanation
The contour (traversed counter-clockwise) crosses the real axis to the left of $-1$ and to the right of $0$ but short of $1$, so it encloses only the pole $z=-1$ and not $z=1$. The residue there is $\frac{2^{-1}}{(-1)-1}=-\frac14$. So the integral is $2\pi i\left(-\frac14\right)=-\frac{i\pi}{2}$, which gives $A=0.5$.