GATE 2026 PH – Question 27

Mathematical Physics · Linear vector spaces: basis, orthogonality and completeness · 1 mark · Multiple select

For which of the following functions does the Laplacian vanish?

$xe^y-ye^x$
$x\cos y-y\cos x$
$e^{x+iy}$
$yx^2-y^3/3-xy$

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Correct answer: (C) $e^{x+iy}$; (D) $yx^2-y^3/3-xy$

Explanation

The Laplacian of a function is $\nabla^2f=\dfrac{\partial^2f}{\partial x^2}+\dfrac{\partial^2f}{\partial y^2}$. A function with $\nabla^2f=0$ is harmonic.

**A. $f=xe^y-ye^x$:** $f_{xx}=-ye^x$, $f_{yy}=xe^y$. The sum $xe^y-ye^x\neq0$. ✗
**B. $f=x\cos y-y\cos x$:** $f_{xx}=y\cos x$, $f_{yy}=-x\cos y$. The sum is not zero. ✗
**C. $f=e^{x+iy}$:** $f_{xx}=e^{x+iy}$, $f_{yy}=i^2e^{x+iy}=-e^{x+iy}$. The sum is 0. ✓ (it is an analytic function of $z=x+iy$)
**D. $f=yx^2-y^3/3-xy$:** $f_{xx}=2y$, $f_{yy}=-2y$. The sum is 0. ✓

Answer **C and D**.