GATE EC 2022 Set 1 — Question 32
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Engineering Mathematics → Complex Analysis → Cauchy's Integral Formula
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Question
A simple closed path in the complex plane is shown in the figure. If where , then the value of is ______ (rounded off to two decimal places).
Correct answer
0.5 to 0.5
Solution
To find the value of , we use Cauchy's Residue Theorem. The integral is given by:The integrand has simple poles at and . From the provided figure, the closed path encloses only the pole at . Next, we determine the orientation of the path . The arrow on the top part of the curve points to the left. For a closed loop in the complex plane, moving left on the upper arc corresponds to a counter-clockwise (positive) orientation. By Cauchy's Residue Theorem, the integral is:The residue at is calculated as:Substituting this back into the integral expression:\oint_C f(z) dz = 2\pi i \left(- rac{1}{4}\right) = -\frac{i\pi}{2}The problem states that the integral is equal to . Comparing the two expressions:Thus, the value of is .
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