Helsinki University of Technology Somersalo/Bingham Mathematics

Mat-1.198 Scattering Theory

1^{st} set of exercises, 29.1.2003

By doing the exercises, you can earn points that will be taken into account when determining your grade for the course: Do these exercises in advance, mark them as solved on the list at the exercise session and be prepared to present your solution on the blackboard.

1. Show that in the two dimensional case, the fundamental solution of the Helmholtz equa-

tion is

Φ(x) = \frac{i}{4} H_{0}^{(1)}(k|x|),

i.e., verify the multiplicative constant. Further, find the explicit form of the asymptotic expansion

Φ(x − y) ∼ \frac{e^{ik|x|}}{\sqrt{|x|}}[f(x̂,y) + O ( \frac{1}{|x|}) ] ,

when |x| → ∞ and |y| < R.

2. Harmonic polynomials in ℝ^{2}: A polynomial P_{n}(x,y) of degree n ∈ ℕ = {0,1,2,...} in ℝ^{2} is homogenous, if

P_{n}(λx,λy) = λ^{n}P_{n}(x,y) for all λ > 0, (x,y) ∈ ℝ^{2}

and harmonic, if

ΔP_{n}(x,y) = 0.

The two-dimensional equivalent to spherical harmonics of ℝ^{3} are defined as the restrictions of the homogenous harmonic polynomials to the unit circle S^{1}, or simply

Y_{n}(θ) = P_{n}(cosθ,sinθ), 0 ≤ θ < 2π.

Show that a basis for these functions is given by {e^{±inθ} }.

3. Verify the recurrence relation

J_{p−1} (z) − J_{p+1} (z) = 2J′_{p}(z).

In particular, deduce the useful formula

J′_{0}(z) = −J_{1}(z).

(Note: This formula holds for the Neumann function N_{p} as well, and thus also for the Hankel functions H_{p}^{(j)} = J_{p}+ (−1)^{j−1} N_{p}, j ∈ {1,2}.)

4. The Hankel transform H_{p} is defined for functions f : ℝ_{+} → ℂ as

H_{p}(f)(k) = ∫_{0}^{∞} J_{p}(kr)f(r)rdr, p = 0,1,...

whenever this integral is convergent. For functions f : ℝ^{2} → ℂ, we define the Fourier transform as

f̂(ξ) = ∫_{ℝ^{2}}e^{−iξ·x}f(x)dx, ξ ∈ ℝ^{2}.

Find an expression for the two-dimensional Fourier transform in terms of the Hankel transforms and the trigonometric Fourier series representation

f(x) = ∑_{m=−∞}^{∞} f_{m}(r)e^{imθ}, x = (r cosθ,r sinθ).

(Assume here that all the necessary integrals and series are convergent, i.e., a formal derivation suffices.)

Useful formulae which you may assume when doing the excercises:

J_{p}(z) = \frac{1}{2π} ∫_{0}^{2π}e^{i(zsinθ−pθ)} dθ = ∑_{j=0}^{∞} \frac{(−1)^{j}}{j!(p + j)!} ( \frac{z}{2})^{p+2j}

The series expansion also holds for p ∈ ℤ_{−} with the following interpretation for j ∈ ℤ_{−}:

\frac{1}{j!} = \frac{j + 1}{(j + 1)!} = \frac{(j + 1)(j + 2)}{(j + 2)!} = ···= \frac{(j + 1)(j + 2)··· (−1)0}{0!} = \frac{0}{1} = 0.

N_{0}(z) = \frac{2}{π} ( ln \frac{z}{2} + γ) + O(z^{2}),  N_{p}(z) = ( \frac{z}{2})^{−p} [−\frac{(p − 1)!}{π} + o(1)] ,  z → 0, Rez > 0

H_{p}^{(j)}(z) = \sqrt{\frac{2}{πz}} e^{i(−1)^{j}[(2p+1) \frac{π}{4}−z]} [1 + O ( \frac{1}{|z|}) ] , |z| → ∞,−π < argz < 2π,j ∈ \{1,2\}

The notation Y_{p} is also often used for the Neumann function N_{p} in literature, for instance in the book by Colton and Kress.