 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2017/4/c/solution

The incident plane wave is periodic in

θ

\sum_{m} R_{m} (r) e^{im θ}

. Substitution into the Helmholtz equation, with

ρ = k r

ρ^{2} R_{m}^{''} + ρ R_{m}^{'} + (ρ^{2} - m^{2}) R_{m} = 0.

Regularity at the origin selects the Bessel function of the first kind, rather than the singular second solution. Thus

e^{ik r c o s θ} = m = - \infty \sum \infty a_{m} J_{m} (k r) e^{im θ} .

a_{m}

are independent of

r

; their values follow from matching near the origin in the next part. The completed identity is the Jacobi-Anger expansion.

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