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GATE2020-ME-2: 27

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The function $f(z)$ of complex variable $z=x+iy$, where $i=\sqrt{-1}$, is given as $f(z)=(x^3-3xy^2)+i \: v(x,y)$. For this function to be analytic, $v(x,y)$ should be

  1. $(3xy^2-y^3) +$ constant
  2. $(3x^2y^2-y^3) +$ constant
  3. $(x^3-3x^2 y) +$ constant
  4. $(3x^2y-y^3) +$ constant

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