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I'd like to determine the error in the asymptotic approximation caused by choosing an incorrect turning point for h(t), as below.

So we consider integrals of the form

f(x)=\int_a^b g(t) e^{x h(t)} . dt,

and we assume a maximum at c, a < c < b. And we consider only real-valued functions and variables at this time.

So normally, you substitute h(t) - h(c) = -s^2.... But what if you choose c incorrectly? It's possible to write h(c+\epsilon) = h(c) + \epsilon^2 h''(c)/2, but then I get the feeling the usage of the Mean-Value theorem (to select -s^2 and not -s) is violated.

Also, this seems to imply that h''(c) must always be known, which isn't the case.

What do people think?

Cheers.