Asymptotic expansion

An asymptotic expansion represents a function by ordered terms whose finite truncations approximate it with progressively smaller errors in a specified limit, even when the infinite series diverges.

Asymptotic scale: An ordered sequence of functions in which each term is negligible relative to its predecessor in a specified limit. Its ordering determines which terms count as successively smaller contributions.
Taylor series: A power series whose coefficients are derivatives of a function at a point. Taylor expansions can be asymptotic near the expansion point, but also have a convergence theory.
Formal power series: A sequence of coefficients treated algebraically as an infinite power series, without requiring convergence. Formal manipulation alone does not establish the remainder estimates of an asymptotic expansion.
Limit of a function: The value a function approaches as its input approaches a specified point or infinity. An asymptotic expansion is defined by behavior in a chosen limiting regime.
Stokes phenomenon: A change in the contribution of exponentially small terms as a complex parameter crosses certain lines. It explains why a single asymptotic description can change across parameter regions.
Big O notation: A notation describing an upper bound on a function's growth or error up to a constant factor. It expresses the remainder bound after truncating an expansion.
Stirling's approximation: An asymptotic formula estimating factorials for large arguments. Its successive correction terms give a classic large-parameter expansion.
Divergent series: A series whose partial sums do not approach a finite limit. Many useful asymptotic expansions diverge while their early truncations remain accurate.
Sequence of functions: An ordered collection of functions indexed by natural numbers or another parameter. Each successive truncation is a function in a sequence of approximations.
Transseries: An expansion combining powers, exponentials, and other scales to describe solutions beyond ordinary power series. It extends asymptotic expansions to include effects invisible at every algebraic order.

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