Steepest-descents/Laplace's method

  • Thread starter highfibre
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So this is a question on the asymptotic evaluation of integrals. It'd help to know the method quite thoroughly....

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.
 

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