GATE 2019 ME (ME2) – Question 14

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

An analytic function of $z=x+iy$ is $f(z)=u(x,y)+iv(x,y)$. Which relations must u and v satisfy?

$u_x=v_y,\ u_y=v_x$
$u_x=v_y,\ u_y=-v_x$
$u_x=-v_y,\ u_y=v_x$
$u_x=-v_y,\ u_y=-v_x$

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Show answer and explanation

Correct answer: (B) $u_x=v_y,\ u_y=-v_x$

Explanation

The Cauchy–Riemann equations are $u_x=v_y$ and $u_y=-v_x$. They equate the complex derivative computed along real and imaginary directions. The minus sign in the second equation is essential. Both are represented by **B**.