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GATE 2016 ME (ME1) – Question 13

Engineering Mathematics · Complex Variables: Analytic functions, Cauchy-Riemann equations, integral theorem, Taylor and Laurent series · 1 mark · Multiple choice

$f(z)=u(x,y)+iv(x,y)$ is analytic in $z=x+iy$, where $i=\sqrt{-1}$. If $u(x,y)=2xy$, then $v(x,y)$ may be ______.

  1. $-x^2+y^2+\text{constant}$
  2. $x^2-y^2+\text{constant}$
  3. $x^2+y^2+\text{constant}$
  4. $-(x^2+y^2)+\text{constant}$

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Correct answer: (A) $-x^2+y^2+\text{constant}$

Explanation

Cauchy–Riemann equations give vy=ux=2y and vx=−uy=−2x. Integrating vy gives v=y²+g(x), and the second equation gives g(x)=−x²+C. Therefore **v=−x²+y²+C, A**. The additive real constant does not affect analyticity.