 Solution

ID: past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2024/paper-3/7/a/solution

ω = 2 π / T

and

q = m + n

. If

q = 0

, the integrand is one and its integral is

T

. If

q

is a nonzero integer,

\int_{0}^{T} e^{iω qt} d t = \frac{e ^{i 2 π q} - 1}{iω q} = 0.

Thus

\int_{0}^{T} e^{iωm t} e^{iωn t} d t = T 1_{m = - n} .

This is Fourier orthogonality in its unconjugated form. With a complex conjugate on the second exponential, the nonzero case would instead be

m = n

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