I like De Bruijn's book, but I like more Wong's, it has everything on asymptotics, applied math in a nutshell :-) But what I like of De Bruijn's book is that you have an explanation in words before going deep into the math. The chapter on the steepest descent method and the solution is very clear using the analogy of a guy going through a valley with hills.

Going back to posted problem, what I have found is (I'm not sure) that you cannot obtain a general solution for general f because the expansion is directly related to the shape of the landscape near the saddle points of f. In these books, the authors explain the case for the natural exponential because we know very well the shape of the landscape, and functions of that kind appear very often.

It seems that this is an open problem.

]]>Thanks for those links. I know the De Bruijn book and am not a big fan of it, despite it's reputation. I have the Wong book but haven't been able to get through it yet.

This site

http://dlmf.nist.gov/2/4/

mentions a few other ideas, that might help or at least point you in a new direction. I am unfamiliar with the Haar method and I haven't read the Olver book so this is a longshot.

Given the following contour integral

where C is a contour, f and g are analytic functions defined over C. What is the asymptotic approximation of the closed-form solution when t->infinity?

In general you cannot use the classical methods from complex analysis like steepest descent or saddle point methods because these methods require

for some h analytic.

I found that there are solutions for general f when t is multiplying f, i.e.,

this is the general steepest descent. There are also solutions when

Working with f, you'll find out that you can't cheat, e.g.,

won't help you. It only works for small values of t.

So, for general f it seems that there is no general solution. What do you think?

For a good reading on asymptotics, you can look at these books from google books

http://books.google.es/books?id=_tnwmvHmVwMC&printsec=frontcover#v=onepage&q=&f=false

http://books.google.es/books?id=xooq99A9anMC&printsec=frontcover#v=onepage&q=&f=false

http://books.google.es/books?id=KQHPHPZs8k4C&printsec=frontcover#v=onepage&q=&f=false

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