 Solution

ID: past-exam-of-the-analysis-course-of-the-university-of-oxford/2009/b/4/iii/solution

\int_{- 1}^{1} P_{n} P_{m} = 2 δ_{nm} / (2 n + 1)

. The allowed exchanges give

\int_{- 1}^{1} \frac{d t}{1 - 2 wt + w ^{2} 1 - 2 z t + z ^{2}} = 2 \sum_{n \geq 0} \frac{( w z ) ^{n}}{2 n + 1} .

Hence it depends only on

r = w z

. Explicitly,

F (r) = 2 \int_{0}^{1} \frac{d s}{1 - r s ^{2}} = ⎩ ⎨ ⎧ 2 artanh (r) / r, 2, 2 arctan (- r) / - r, r > 0, r = 0, r < 0.

The series form alone already proves the required dependence.

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