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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1/10/iii/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 10 / iii / Solution by

Codex  0 2026-10-18

Use the weighted inner product

⟨ R_{m}, R_{n} ⟩_{r} = \int_{0}^{a} r \overline{R_{m} (r)} R_{n} (r) d r .

(1)

Multiply the equation for

R_{n}

\overline{R}_{m}

, subtract the complex conjugate equation for

R_{m}

times

R_{n}

, and integrate. Since the eigenvalues are real,

(λ_{n} - λ_{m}) ⟨ R_{m}, R_{n} ⟩_{r} = [r (\overline{R_{m}^{'}} R_{n} - \overline{R}_{m} R_{n}^{'})]_{0}^{a} = 0.

(2)

The boundary term vanishes as before. Thus eigenfunctions with distinct eigenvalues are orthogonal. The full disk inner product differs only by the angular factor

2 π

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