The angular distribution of intensity scattered by a certain process is measured as a…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Special functions and orthogonal polynomials · Function spaces and operator theory

Conditions of Validity, Ch Eigenfunction expansions, convergence and the Rayleigh quotient

Problem

The angular distribution of intensity scattered by a certain process is measured as a function of μ=cos⁡θ\mu=\cos\theta and found to be f(μ)=∣μ∣f(\mu)=|\mu| for −1≤μ≤1-1\le\mu\le1 (dimensionless). It is to be expanded in Legendre polynomials Pℓ(μ)P_\ell(\mu), the eigenfunctions of −[(1−μ2)y′]′=ℓ(ℓ+1)y-\big[(1-\mu^2)y'\big]'=\ell(\ell+1)y on [−1,1][-1,1] with weight w=1w=1 and normalisation Pℓ(1)=1P_\ell(1)=1, for which ∥Pℓ∥w2=2/(2ℓ+1)\|P_\ell\|_w^2=2/(2\ell+1). Compute the first three coefficients c0c_0, c1c_1, c2c_2 in f(μ)=∑ℓ=0∞cℓPℓ(μ)f(\mu)=\sum_{\ell=0}^\infty c_\ell P_\ell(\mu), using cℓ=⟨Pℓ,f⟩w/∥Pℓ∥w2c_\ell=\langle P_\ell,f\rangle_w/\|P_\ell\|_w^2.

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