Past exam of the theoretical physics II course of the University of Cambridge / 2016 / Paper 1 / 1 / c

Table of contents
- Solution c

Solution

For a nonnegative integer

m

, expand the plane wave in powers of

ρ

:

e^{i ρ c o s θ} = \sum_{n = 0}^{\infty} \frac{( i ρ ) ^{n}}{n !} (\frac{e ^{i θ} + e ^{- i θ}}{2})^{n} .

The first contribution to the positive angular harmonic

e^{im θ}

occurs at

n = m

. Its coefficient is

(i ρ)^{m} / (2^{m} m!)

. Comparing this with

a_{m} J_{m} (ρ) \sim a_{m} ρ^{m} / (2^{m} m!)

gives

a_{m} = i^{m}

for

m \geq 0

.

The plane wave is even in

θ

, so its

m

and

- m

Fourier coefficients agree. Therefore

a_{- m} J_{- m} = a_{m} J_{m}

, implying

a_{- m} = (- 1)^{m} i^{m} = i^{- m}

. Hence

a_{m} = i^{m} (m \in Z), e^{i ρ c o s θ} = m = - \infty \sum \infty i^{m} J_{m} (ρ) e^{im θ} .

This is the cosine form of the Jacobi-Anger expansion. The factorial asymptotic stated in the source is used for nonnegative order; applying it literally to negative integers would introduce an undefined factorial. The Integer-order Bessel reflection identity is the required extension, not a separate set of divergent radial solutions.

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