textbooks on asymptotic expansions

De Bruijn's "Asymptotic methods in analysis" is an excellent book for beginners. You'll need to work through it diligently to learn everything but no advanced a priori knowledge is required. Also, you can easily download it from many online places that do not worry too much about copyright and, even if you decide to stay law-abiding, it goes for under ten bucks on amazon.

There is a very large literature on asymptotic expansions, including books. What is the best books depends on your needs. A comprehensive (advanced) book oriented at physicists and applied mathematicians is

MR0499926 Dingle, R. B. Asymptotic expansions: their derivation and interpretation. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1973.

Another good book for physicists/engineers is

MR1721985 Bender, Carl M.; Orszag, Steven A. Advanced mathematical methods for scientists and engineers. I. Asymptotic methods and perturbation theory. Springer-Verlag, New York, 1999.

Books more oriented at pure mathematicians is

MR0435697 Olver, F. W. J. Asymptotics and special functions. Computer Science and Applied Mathematics. Academic Press, New York-London, 1974.

and (an older book)

MR0115035 Ford, Walter B. Studies on divergent series and summability & The asymptotic developments of functions defined by Maclaurin series. Chelsea Publishing Co., New York 1960 x+342

http://www.amazon.com/Asymptotics-Summability-Monographs-Surveys-Mathematics/dp/1420070312/

Asymptotics and Borel Summability, O. Costin