Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 14A (Methods)

Apply separation of variables to the Laplace equation in polar coordinates by writing $\varphi (r,\theta )=R(r)\Theta (\theta )$. Division by $R\Theta$ givesThe requirement that $\Theta$ be a real $2\pi$-periodic function restricts the separation constants to $n^2$, with angular factors $\cos (n\theta )$ and $\sin (n\theta )$. For $n\ge 1$, the radial ordinary differential equation is an Euler equation with solutions $r^n$ and $r^{-n}$. For the zero mode, $\Theta$ is constant andso $R=a_0+c_0\log r$. By linearity, superposition givesThis is the separated expansion of a harmonic function in a circular region.

Regularity at the origin excludes the natural logarithm and every negative radial power. ThusThe coefficients $a_0,a_n,b_n$ remain arbitrary.

Regularity at infinity excludes the natural logarithm and every positive radial power. ThusThe constant $a_0$ and the decaying coefficients $c_n,d_n$ remain arbitrary.

An annulus stays away from both the origin and infinity, so every displayed radial mode is regular there. Therefore none of the coefficients is forced to vanish.

Write $s=\log r$, so the two circular boundaries are $s=0$ and $s=2$. The zero angular mode must interpolate between $-1$ and $1$, giving $s-1=\log r-1$.

For the $n$th cosine mode, the radial factor has equal value $A_n$ at both boundaries. The unique harmonic function with those data isHence the solution of the annular Dirichlet problem isAt $r=1$ and $r=e^2$, the hyperbolic cosine quotient equals one, so the boundary conditions are satisfied term by term.

The mean of $f$ is zero. Its Fourier cosine series coefficients areThus $A_n=0$ for even $n$ and $A_n=4/(\pi n^2)$ for odd $n$. Substitution into part (i) yields