Asymptotic expansions

In mathematics, an asymptotic expansion, asymptotic series, or Poincaré expansion is a formal series used to approximate a given function near a specific point or at infinity. The term asymptotic series is commonly reserved for a divergent series whose terms initially decrease to a minimum magnitude and then increase without bound. Although divergent, the truncated series provides an approximation with increasing accuracy as the function's argument approaches the asymptotic limit. These expansions are useful when a standard Taylor series converges too slowly, as an asymptotic series can often give high accuracy with only a few terms. An asymptotic expansion depends on the domain and limit point. A function of a real variable may require distinct series near the origin versus at infinity. An entire function of a complex variable may require distinct asymptotic series for different sectors of the complex plane, a behavior called the Stokes phenomenon. The most common type of asymptotic expansion is a power series in either positive or negative powers. Asymptotic series commonly occur when using the Euler–Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often generate an asymptotic expansion. Asymptotic analysis plays an important role in singular perturbation theory and for solving nonlinear equations in fluid mechanics. Today, asymptotic series are used in the analysis of algorithms in computer science, the evaluation of complex integrals and partial sums, and the study of differential equations and difference equations.