Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 14A / a

Table of contents
- Solution a
- i a
  - Solution i
- ii a
  - Solution ii
- iii a
  - Solution iii

Solution

Apply separation of variables to the Laplace equation in polar coordinates by writing

ϕ (r, θ) = R (r) Θ (θ)

R Θ

gives

\frac{r ( r R ^{'} ) ^{'}}{R} = - \frac{Θ ^{''}}{Θ} = n^{2} .

The requirement that

Θ

be a real

2 π

-periodic function restricts the separation constants to

n^{2}

, with angular factors

cos (n θ)

and

sin (n θ)

. For

n \geq 1

, the radial ordinary differential equation is an Euler equation with solutions

r^{n}

and

r^{- n}

. For the zero mode,

Θ

is constant and

(r R^{'})^{'} = 0,

so

R = a_{0} + c_{0} lo g r

. By linearity, superposition gives

ϕ (r, θ) = a_{0} + c_{0} lo g r + n = 1 \sum \infty (a_{n} r^{n} + c_{n} r^{- n}) cos (n θ) + n = 1 \sum \infty (b_{n} r^{n} + d_{n} r^{- n}) sin (n θ) .

This is the separated expansion of a harmonic function in a circular region.

i

Solution

Regularity at the origin excludes the natural logarithm and every negative radial power. Thus

c_{0} = 0, c_{n} = d_{n} = 0 (n \geq 1) .

The coefficients

a_{0}, a_{n}, b_{n}

remain arbitrary.

ii

Solution

Regularity at infinity excludes the natural logarithm and every positive radial power. Thus

c_{0} = 0, a_{n} = b_{n} = 0 (n \geq 1) .

The constant

a_{0}

and the decaying coefficients

c_{n}, d_{n}

remain arbitrary.

iii

Solution

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.

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