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GATE 2019 EE – Question 14

Engineering Mathematics · Complex Variables: Analytic functions, Cauchy integral formula, Taylor series, Laurent series, Residue theorem · 1 mark · Multiple choice

Which one of the following functions is analytic in the region $|z|\le1$?

  1. $\dfrac{z^2-1}{z}$
  2. $\dfrac{z^2-1}{z+2}$
  3. $\dfrac{z^2-1}{z-0.5}$
  4. $\dfrac{z^2-1}{z+j0.5}$

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Correct answer: (B) $\dfrac{z^2-1}{z+2}$

Explanation

A rational function is analytic wherever its denominator is not zero. The poles are at $z=0$ (A), $z=-2$ (B), $z=0.5$ (C) and $z=-j0.5$ (D). Only the pole in B lies outside $|z|\le1$.