The GATE Grind

GATE 2017 ME (ME2) – Question 39

Engineering Mathematics · Differential Equations: Initial and boundary value problems, Laplace transforms · 2 marks · Multiple choice

If $f(z)=(x^2+ay^2)+ibxy$ is analytic in $z=x+iy$, where $i=\sqrt{-1}$, then ______.

  1. a = −1, b = −1
  2. a = −1, b = 2
  3. a = 1, b = 2
  4. a = 2, b = 2

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) a = −1, b = 2

Explanation

Put u=x²+ay² and v=bxy. Cauchy–Riemann equations require ux=vy and uy=−vx, hence2x=bx and2ay=−by for all x,y. Therefore b=2 and a=−1, giving **B**; the function then equals z².