GATE 2024 PI – Question 27
$f(x,y)=u(x,y)+iv(x,y)$ is analytic, with real x,y,u,v. Which ONE OR MORE equations hold?
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Correct answer: (A) $u_{xx}+u_{yy}=0$; (B) $v_{xx}+v_{yy}=0$; (D) $u_xv_x+u_yv_y=0$
Explanation
The Cauchy–Riemann equations $u_x=v_y$, $u_y=-v_x$ imply u and v are harmonic, giving A and B. They also give $u_xv_x+u_yv_y=u_x(-u_y)+u_yu_x=0$, giving D. C need not hold; for $f(z)=z^2$, $u_{xx}=2$ and $v_{yy}=0$. **A,B,D**.