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I have been reading Semi-classical analysis by Guillemin and Sternberg. At the end of Chapter 8, they gave an abstract version of the stationary phase method. I have a hard time figuring out what $a_i$ in the asymptotic expansion of theorem 8.14.1 are. Any suggestions are welcomed.

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the $a_i$'s are complex coefficients in the asymptotic expansion $a(h)=\sum_{i=0}^\infty a_i h^i$; a more explicit expression is given on page 275-276 of Guillemin and Sternberg:

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