For f(z)=x3-3xy2+iv(x,y), with z=x+iy and i=-1, which v makes f analytic?

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

GATE 2020 ME (ME2) – Question 37

Engineering Mathematics · Complex Variables: Analytic functions, Cauchy-Riemann equations, integral theorem, Taylor and Laurent series · 2 marks · Multiple choice

For $f(z)=x^3-3xy^2+iv(x,y)$, with $z=x+iy$ and $i=\sqrt{-1}$, which v makes f analytic?

$(3xy^2-y^3)+\text{constant}$
$(3x^2y^2-y^3)+\text{constant}$
$(x^3-3x^2y)+\text{constant}$
$(3x^2y-y^3)+\text{constant}$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (D) $(3x^2y-y^3)+\text{constant}$

Explanation

Cauchy–Riemann gives vy=3x²−3y² and vx=6xy. Integrate the first equation: v=3x²y−y³+g(x). The second implies g′=0. Thus **v=3x²y−y³+constant,D**, consistent with f=z³+iC.