Past exam of the year 1 of the computer science course of the University of Oxford / 2005 / Calculus, linear algebra, discrete mathematics, logic and proof, and models of computation / 4 / a / i / Solution

For $r>0$, the chain rule gives $r_x=x/r$ and $r_y=y/r$. Applying it twice to the radial function $u=f(r)$ givesUsing $x^2+y^2=r^2$, Laplace's equation becomesTwo integrations now giveThese are the general solutions of the specified radial function form on an annulus or a punctured plane. If the solution must extend continuously to the origin, the logarithm term forces $A=0$, leaving only a constant function.