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GATE 2021 ME (ME1) – Question 37

Engineering Mathematics · Complex Variables: Analytic functions, Cauchy-Riemann equations, integral theorem, Taylor and Laurent series · 2 marks · Multiple choice

Let $C$ represent the unit circle centered at origin in the complex plane, and complex variable, $z=x+iy$. The value of the contour integral $\oint_C\frac{\cosh3z}{2z}\,dz$ (where integration is taken counter clockwise) is

  1. 0
  2. 2
  3. $\pi i$
  4. $2\pi i$

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Show answer and explanation

Correct answer: (C) $\pi i$

Explanation

The numerator is analytic and the only pole inside C is z=0. Its residue is cosh(0)/2=1/2.

The residue theorem gives $2\pi i(1/2)=\mathbf{\pi i}$, **C**. Counterclockwise orientation gives the positive sign.