Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 30D / b / iii

Table of contents
- Solution iii

Solution

At the two saddles,

\frac{z _{\pm}^{2} - 1}{z _{\pm} - t} = 2 z_{\pm} = 2 e^{\pm i θ}, f (z_{\pm}, t) = \frac{1}{z _{\pm} - t} = \mp \frac{i}{s i n θ},

and

∣ ϕ^{''} (z_{\pm}) ∣ = \frac{1}{s i n θ} .

Applying the method of steepest descent with the oriented angles from part (ii), and including the prefactor

1/ (2^{n + 1} πi)

, gives two conjugate contributions. If

Φ = (n + \frac{1}{2}) θ - \frac{π}{4},

their sum is

\frac{1}{2 π} \frac{2 π}{n s i n θ} (e^{i Φ} + e^{- i Φ}) .

Therefore the Debye asymptotic for Legendre polynomials is

P_{n} (cos θ) \sim \frac{2}{πn sin θ} cos (n θ + \frac{θ}{2} - \frac{π}{4}) .

Thus

A (θ) = \frac{2}{πn sin θ} .

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