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

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.

Use the product rule for the Laplace operator:Thus Laplace's equation, away from the origin, reduces to $g^{\prime \prime }+3g^{\prime }/r=0$. On the line $x=0$ the displayed expression vanishes automatically; requiring the equation on an open region gives the same radial equation by continuity from $x\ne 0$. Multiplying by $r^3$ gives $(r^3{g}^{\prime }{)}^{\prime }=0$, so $g^{\prime }=C{r}^{-3}$ and $g=A+B{r}^{-2}$. ThereforeIf extension through the origin is required, $B=0$: along the horizontal axis the second term is $B/x$. The remaining harmonic function is $Ax$.