Robin Pemantle and Mark C. Wilson, to appear in SIAM Review, June 2008.

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Abstract: Let $latex F$ be a power series in at least two variables that defines a meromorphic function in a neighbourhood of the origin; for example, $latex F$ may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of $latex F$, uniform in certain explicitly defined cones of directions.

The purpose of this article is to illustrate the use of these techniques on a variety of problems of combinatorial interest. The first part reviews the Morse-theoretic underpinnings of these techniques, and then summarizes the necessary results so that only elementary analyses are needed to check hypotheses and carry out computations. The remainder focuses on combinatorial applications. Specific examples deal with enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, descents and solutions to integer equations. After the individual examples, we discuss three broad classes of examples, namely functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. Generating functions derived in these three ways are amenable to our asymptotic analyses, and we state some further general results that apply to these cases.