Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 30A / a / Solution

Near the nondegenerate stationary point, write

g (t) = g (t_{0}) + g^{''} (t_{0}) (t - t_{0})^{2} /2 + O ((t - t_{0})^{3})

and replace

f (t)

by

f (t_{0})

to leading order. Scaling

t - t_{0}

by

λ^{- 1/2}

reduces the local integral to the Fresnel integral, while the remaining nonstationary region contributes

O (λ^{- 1})

. The stationary phase result is

I (λ) = f (t_{0}) \frac{2 π}{λ ∣ g ^{''} ( t _{0} ) ∣} exp (iλ g (t_{0}) + \frac{iπ}{4} sgn g^{''} (t_{0})) + O (λ^{- 1}) .

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