 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1/8/e/solution

Since

P_{0} = 1

, part (d) gives

Q_{0}^{'} = A_{0} / (1 - x^{2})

- 1 < x < 1

Q_{0} (x) = \frac{A _{0}}{2} lo g \frac{1 + x}{1 - x} + B .

The condition at zero fixes

B = 0

. Thus the full conclusion from the stated condition is

Q_{0} (x) = A_{0} arctanh x = \frac{A _{0}}{2} lo g \frac{1 + x}{1 - x}, A_{0} \neq = 0.

That condition alone does not fix the multiplicative constant. With the usual Legendre function of the second kind normalization

Q_{0}^{'} (0) = 1

, one has

A_{0} = 1

. On real exterior intervals replace the logarithm by

lo g ∣ (1 + x) / (1 - x) ∣

; complex continuation requires a choice of logarithm branch.

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