 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-2/18/b/ii/solution

The analogous orthogonal projection onto the cosine terms gives

a_{n} = \frac{1}{L} \int_{- L}^{L} f (x) cos (nπ x / L) d x, n \geq 0 .

For

n = 0

, this means

a_{0} = L^{- 1} \int_{- L}^{L} f (x) d x

; the actual constant in the Fourier series is

a_{0} /2

because the constant function has squared norm

2 L

 New to topics? Read the docs here!