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Asymptotic Series / Poincaré Expansion: Simple Definition, Example

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An asymptotic series (also called an asymptotic expansion or Poincaré expansion) is a way of investigating the general behavior of a function as the independent variable (e.g. x) becomes very large. The procedure, which truncates a function after a certain number of terms (a method of partial sums), can get very good approximations with relatively few calculations—especially when the calculations are so complex it would overwhelm most computers (Beckenbach, 2013).

One of the simplest ways to get an asymptotic series is to do a change of variable x→ 1/x, then perform a series expansion about zero. Other ways to generate these series includes the Euler–Maclaurin summation formula, Integral transforms (e.g. the Laplace transform and Mellin transform) and repeated integration by parts.

As an example, the following expansion can be used to find an approzimation for the Riemann zeta function:

Any function smaller that every term in an asymptotic series is subdominant to the series. This means that there isn’t a unique function associated with any particular asymptotic series, but rather one of these series defines an entire class of asymptotically equivalent functions. Theses different asymptotic sequences are called gauges (Paulsen, 2013; Goldstein, 2020).


Convergence & Divergence of Asymptotic Series

“One remarkable fact of applied mathematics is the ubiquitous appearance of divergent series, hypocritically renamed asymptotic expansions….The challenge of explaining what an asymptotic expansion is ranks among the outstanding taboo problems of mathematics. —Gian-Carlo Rota, in Indiscrete Thoughts (1997), p. 222″

If you find the formal definitions of asymptotic theory challenging to wrap your head around, you aren’t alone in your confusion. Several prominent authors use the term “asymptotic series” to (incorrectly) describe divergent series (e.g. Dingle, 2013; Gradshteyn & Ryzhik, 2007). In general, while all divergent series are asymptotic series (Cousteix & Mauss, 2007; Erdélyi, 1987), the converse isn’t necessarily true: Asymptotic expansions may or may not be divergent. Therefore, you shouldn’t worry about whether or not a particular asymptotic series will converge or diverge until after you have performed the expansion (Michon, 2020).

References

Beckenbach, E. (2013). Modern Mathematics for the Engineer: Second Series. Dover Publications.
Cousteix, J. & Mauss, J. (2017). Asymptotic Analysis and Boundary Layers. Springer.
Dingle, R. (2013). Asymptotic Expansions: Their Derivation and Interpretation. Academic Press.
Erdélyi, A. (1987). Asymptotic Expansions. New York: Dover,.
Goldstein, R. (2020). Asymptotic versus Convergent series. Optimal truncation. Retrieved November 22, 2020 from: http://www.damtp.cam.ac.uk/user/gold/pdfs/teaching/L3.pdf
Gradshteyn, S. and Ryzhik, I. (2007). Table of Integrals, Series, and Products. Edited by A. Jeffrey and D. Zwillinger. Academic Press, New York, 7th edition.
Michon, G. (2020). Asymptotic Analysis. Retrieved November 20, 2020 from: http://www.numericana.com/answer/asymptotic.htm#series
Paulsen, W. (2013). Asymptotic Analysis and Perturbation Theory. Chapman and Hall/CRC.
Rota, G. (2000). Indiscrete Thoughts. Birkhäuser.

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Stephanie Glen. "Asymptotic Series / Poincaré Expansion: Simple Definition, Example" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/asymptotic-series-poincare-expansion/
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