I have the following problem for which I cannot find an answer. I posted this same problem in "physicsforums" and "mathhelpforum" and still nobody was able to give me an anwser. I looked everywhere in the internet (journal, books, etc) and also found nothing. Maybe some of you guys can give me some hints.

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