GATE 2019 ME (ME1) – Question 37

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

A harmonic function satisfies Laplace’s equation. For $u(x,y)=2x^2-2y^2+4xy$, find its conjugate harmonic function v(x,y).

$4xy-2x^2+2y^2+C$
$4y^2-4xy+C$
$2x^2-2y^2+xy+C$
$-4xy+2y^2-2x^2+C$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $4xy-2x^2+2y^2+C$

Explanation

Cauchy–Riemann equations give $v_y=4x+4y$ and $v_x=4y-4x$. Integration yields $v=4xy+2y^2+g(x)$, with $g^{\prime}(x)=-4x$. Thus **$v=4xy-2x^2+2y^2+C$, A**.