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GATE 2015 EC – Question 14

Engineering Mathematics · Complex Analysis · 1 mark · Multiple choice

Let $z = x + iy$ be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction. Which one of the following statements is NOT TRUE?

  1. The residue of $\frac{z}{z^2 - 1}$ at $z = 1$ is 1/2
  2. $\oint_C z^2 dz = 0$
  3. $\frac{1}{2\pi i}\oint_C \frac{1}{z} dz = 1$
  4. $\bar{z}$ (complex conjugate of $z$) is an analytical function

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Correct answer: (D) $\bar{z}$ (complex conjugate of $z$) is an analytical function

Explanation

The residue at $z = 1$ is $\frac{1}{1 + 1} = \frac{1}{2}$ (A is true), $z^2$ is analytic so its closed contour integral is 0 (B is true), and the integral of $\frac{1}{z}$ gives $2\pi i$ (C is true). The conjugate $\bar{z}$ does not satisfy the Cauchy-Riemann equations, so it is not analytic.