GATE ME 2025 Set 1 — Question 45
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Engineering Mathematics → Complex Variables → Cauchy's Integral Theorem
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Question
If is the unit circle in the complex plane with its center at the origin, then the value of in the equation given below is __________ (rounded off to 1 decimal place).
Correct answer
0.1 to 0.1
Solution
The given integral is , where is the unit circle .The singularities of the integrand are the roots of the denominator:
Since the contour is the unit circle , all singularities lie outside the contour ().According to Cauchy's Integral Theorem, if a function is analytic within and on a simple closed contour , then .Thus, .Comparing this with the given form :
.Rounding to one decimal place, the value is . The official answer range is to .
1.
2.
The magnitudes of these singularities are and . Since the contour is the unit circle , all singularities lie outside the contour ().According to Cauchy's Integral Theorem, if a function is analytic within and on a simple closed contour , then .Thus, .Comparing this with the given form :
.Rounding to one decimal place, the value is . The official answer range is to .
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