 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-2/1/b/iii/solution

\int_{- 1}^{1} \frac{y _{m} ( x ) y _{n} ( x )}{1 - x ^{2}} d x = \frac{2}{π} \int_{0}^{π} sin (m θ) sin (n θ) d θ .

m, n

, the product-to-sum identity gives

\frac{1}{π} \int_{0}^{π} [cos ((m - n) θ) - cos ((m + n) θ)] d θ = 0,

because each integrated sine has integer-multiple-of-

π

endpoints. For

m = n

, the integral is one by the normalization already calculated. Hence

\int_{- 1}^{1} w (x) y_{m} (x) y_{n} (x) d x = δ_{mn} .

This verifies orthonormality explicitly, with

δ_{mn}

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