ABSTRACT

We can deduce very precise asymptotic estimates about counting sequences by viewing formal series as analytic functions. The growth of the coefficients of a series is directly related to how the function behaves at its singularities. To understand the connection, we first discuss convergence of formal power series, and then consider the implications of writing coefficients as complex contour integrals. Estimating contour integrals is the centerpiece of analytic combinatorics. The fundamentals that we review here form the basis of the multivariable case. Thus, even if you are familiar with these results, it is useful to have them in mind as a foundation for the more general setting.