 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-2/18/b/ii/solution

ℓ = m L

m > 0

, then

k_{n} ℓ = 2 πnm

. All sine phase factors vanish and all cosine phase factors equal one. Therefore

g (x) = f (x)

for the periodic extension, and every Fourier coefficient is unchanged.

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