GATE 2022 EC Q.32: solution and answer

GATE 2022 · ECQ.32 · 1 mark · NAT

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

\oint_{C} \frac{2 ^{z}}{z ^{2} - 1} d z = - iπ A,

where $i = - 1$ , then the value of $A$ is ______ (rounded off to two decimal places).

GATE 2022 EC question 32: Complex plane with horizontal axis labelled Re(z) and vertical axis labelled Im(z). — Figure from the GATE 2022 EC paper. It is part of the question.

Answer and solution

ANSWER

$A = 0.50$

Official answer: 0.50Source: Official answer key, GATE 2022 EC (IIT Kharagpur)

Singularities. $z^{2} - 1 = (z - 1) (z + 1)$ , so the integrand has simple poles at $z = 1$ and $z = - 1$ . The numerator $2^{z} = e^{z l n 2}$ is entire and not zero at either point, so there are no other singularities.
Read the figure. $C$ meets the real axis at about $Re (z) \approx - 1.7$ on the left and $Re (z) \approx 0.6$ on the right. The right crossing is between the origin and the tick at $(1, 0)$ . So $z = - 1$ is inside $C$ and $z = 1$ is outside. The arrows point left on the top arc and up on the lower-right arc, so $C$ is traversed counterclockwise (the positive sense).
Residue at the enclosed pole. For a simple pole, $Res_{z = z_{0}} f = lim_{z \to z_{0}} (z - z_{0}) f (z)$ : $Res_{z = - 1} \frac{2 ^{z}}{( z - 1 ) ( z + 1 )} = \frac{2 ^{z}}{z - 1}_{z = - 1} = \frac{2 ^{- 1}}{- 2} = - \frac{1}{4}$
Cauchy's residue theorem. The contour is counterclockwise and encloses only $z = - 1$ : $\oint_{C} \frac{2 ^{z}}{z ^{2} - 1} d z = 2 π i (- \frac{1}{4}) = - \frac{iπ}{2}$
Match with the given form. $- iπ A = - \frac{iπ}{2}$ , so $A = \frac{1}{2} = 0.50$ .

KEY IDEA

Only the poles inside the contour count. The figure puts $z = - 1$ inside $C$ and $z = 1$ outside, so the integral is $2 π i$ times the single residue $Res_{z = - 1} = \frac{2 ^{- 1}}{- 1 - 1} = - \frac{1}{4}$ . That gives $- \frac{iπ}{2}$ , so $A = \frac{1}{2}$ .

WATCH OUT FOR

Counting both poles. The residue at $z = 1$ is $\frac{2 ^{1}}{2} = 1$ , and adding it gives $2 π i \cdot \frac{3}{4} = \frac{3 π i}{2}$ , so $A = - 1.50$ . But the contour crosses the real axis at about $0.6$ , to the left of $(1, 0)$ , so $z = 1$ is outside. Two other slips give $A = - 0.50$ : reading the orientation as clockwise, or dropping the sign of $z - 1 = - 2$ at $z = - 1$ .

From the GATE 2022 EC paper. Every question of this paper · Download the full paper · All 24 Complex analysis questions, with notes

GATE 2022 EC Q.32: solution and answer

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