 Solution

ID: past-exam-of-the-perturbation-methods-and-applied-complex-variables-course-of-the-university-of-oxford/2009/a/1/ii/solution

A saddle point of the exponential phase occurs where

(i t^{3})^{'} = 3 i t^{2} = 0

. Hence the only saddle is

t = 0.

It is degenerate: the second phase derivative vanishes there, while the third is

6 i \neq = 0

. This cubic degeneracy produces the

x^{- 1/3}

scale. The other endpoint

t = 1

is not a saddle, since the phase derivative there is

3 i

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