Meromorphic functions If h(z) = \frac{f(z)}{g(z)} is a
meromorphic function and a is its
pole closest to the origin with order m, then :[z^n] h(z) \sim \frac{(-1)^m m f(a)}{a^m g^{(m)}(a)} \left( \frac{1}{a} \right)^n n^{m-1} \quad as n \to \infty
Tauberian theorem If :f(z) \sim \frac{1}{(1 - z)^\sigma} L(\frac{1}{1 - z}) \quad as z \to 1 where \sigma > 0 and L is a
slowly varying function, then :[z^n]f(z) \sim \frac{n^{\sigma-1}}{\Gamma(\sigma)} L(n) \quad as n \to \infty See also the
Hardy–Littlewood Tauberian theorem.
Circle Method For generating functions with
logarithms or
roots, which have
branch singularities.
Darboux's method If we have a function (1 - z)^\beta f(z) where \beta \notin \{0, 1, 2, \ldots\} and f(z) has a
radius of convergence greater than 1 and a
Taylor expansion near 1 of \sum_{j\geq0} f_j (1 - z)^j, then :[z^n](1 - z)^\beta f(z) = \sum_{j=0}^m f_j \frac{n^{-\beta-j-1}}{\Gamma(-\beta-j)} + O(n^{-m-\beta-2}) See
Szegő (1975) for a similar theorem dealing with multiple singularities.
Singularity analysis If f(z) has a singularity at \zeta and :f(z) \sim \left(1 - \frac{z}{\zeta}\right)^\alpha \left(\frac{1}{\frac{z}{\zeta}}\log\frac{1}{1 - \frac{z}{\zeta}}\right)^\gamma \left(\frac{1}{\frac{z}{\zeta}}\log\left(\frac{1}{\frac{z}{\zeta}}\log\frac{1}{1 - \frac{z}{\zeta}}\right)\right)^\delta \quad as z \to \zeta where \alpha \notin \{0, 1, 2, \cdots\}, \gamma, \delta \notin \{1, 2, \cdots\} then :[z^n]f(z) \sim \zeta^{-n} \frac{n^{-\alpha-1}}{\Gamma(-\alpha)} (\log n)^\gamma (\log\log n)^\delta \quad as n \to \infty
Saddle-point method For generating functions including
entire functions. Intuitively, the biggest contribution to the
contour integral is around the
saddle point and estimating near the saddle-point gives us an estimate for the whole contour. If F(z) is an admissible function, then :[z^n] F(z) \sim \frac{F(\zeta)}{\zeta^{n+1} \sqrt{2 \pi f^{''}(\zeta)}} \quad as n \to \infty where F^'(\zeta) = 0. See also the
method of steepest descent. == Notes ==