Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2002 / Paper 1 / 12 / i / Solution

The zero value at $x=0$ removes the even factor in $x$, and the zero value at $y=0$ removes the even factor in $y$. Up to multiplicative constants, the nonzero separated modes areThe zero-separation case also supplies $\boxed{xy}$. Each is a harmonic function and vanishes on both axes.