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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2/18/d/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 18 / d / Solution by

Codex  0 2026-10-18

For a real

2 L

-periodic square-integrable function with the convention

f \sim a_{0} /2 + \sum_{n \geq 1} (a_{n} cos (nπ x / L) + b_{n} sin (nπ x / L))

, Parseval's identity is

\frac{1}{L} \int_{- L}^{L} ∣ f (x) ∣^{2} d x = \frac{a _{0}^{2}}{2} + n = 1 \sum \infty (a_{n}^{2} + b_{n}^{2}) .

(1)

It states that the squared L2 norm equals the sum of the squared components in the orthogonal Fourier basis, with the normalization factors shown. In the present even-function case every

b_{n}

vanishes.

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