Related Links (Outside this Site)

Asymptotic Analysis by Adolf J. Hildebrand (UIUC, Math 595, Fall 2009).
Complex Variables, Contour Integration by Joceline Lega (1998).
Padé and Algebraic Approximants applied to the Quantum Anharmonic Oscillator (pdf)
by Christopher Orth, University of Minnesota, Morris (2005年12月12日)
Adding, Multiplying and the Mellin Transform by Greg Muller (2007).

The n-category Café (discussing papers by Tom Leinster, in 2006 & 2007):
Euler characteristic of a category (2006年10月11日)
Euler characteristic of a category as the sum of a divergent series (2007年07月09日)

This Week's Finds in Mathematical Physics by John C. Baez : 124 | 125 | 126 (1998)
Euler's Proof that z(-1) = -1/12 (2003) | The Number 24 (2008)

Open Problems in Asymptotics Relevant to Sequence Transformations with Special Emphasis to the Summation of Divergent Stieltjes Series
by Ernst Joachim Weniger, Universität Regensburg (1995年03月15日).

Euler-Maclaurin formula, Bernoulli numbers, Zeta function & analytic continuation by Terry Tao (2010).

Divergence of perturbation theory: Steps towards a convergent series.
by Gerardo Enrique Oleaga Apadula & Sergio A. Pernice (1998)
[ On the applicability of Lebesgue's dominated convergence theorem ]

Wikipedia : Asymptotic analysis | Asymptotic expansions | Big O notation | Euler-Maclaurin formula
Perturbation theory | Watson's lemma (1918) | WKB approximation (Jeffreys, 1923)

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(2012年08月01日) Fundamentals of Asymptotics
Only zero is asymptotic to zero.

Let's first consider numerical functions (where division makes sense):

Numerical Functions :

Two numerical functions f and g are called asymptotic (or equivalent) to each other in the neighborhood of some limit point L (possibly at infinity) when the ratio f (x) / g (x) tends to 1 as x tends to L. In other words, the following two notations are equivalent, by definition:

f (x) ~ g (x) ( x ® L ) or f (x) / g (x) ® 1

x ® L

Likewise, the statement " f (x) is negligible  compared to g (x) as x tends to L " is denoted or defined as follows:

f (x) << g (x) ( x ® L ) or f (x) / g (x) ® 0

x ® L

In the US, this is sometimes read f (x) is a lot less than g (x) (as x tends to L). That's misleading because the relation is unrelated to ordering in the real line. For example, both of the following relations hold as x tends to 0:

-1 < x² but x² << -1

You can manipulate algebraically an asymptotic equivalence exactly as you would an ordinary equation, except that you're not allowed to transpose everything to one side of the equation! Nothing (but zero itself) is asymptotic to zero...

Extension to Vectorial Functions :

For vectorial functions, the symmetry in the above definitions must be broken. Negligibility is not difficult to define in a normed vector space: One quantity is negligible compared to another when the norm of the first is negligible compared to the norm of the other. With this in mind, we can promote to a definition among vectors what's a simple characteristic theorem for equivalent scalars quantities (with the definitions given above):

Definitions for Vectorial Asymptotics

Is zero asymptotic to zero?

In asymptotics, "zero" is any function which is identically equal to 0 (the null vector) in some neighborhood of the relevant limit point. The following relations are valid whenever f is a nonzero quantity:

0 << f and f ~ f

By convention, we retain the validity of those two for zero quantities: Only zero is negligible compared to zero. Only zero is equivalent to zero.

Schwartz functions | Asymptotic approximations | Wikipedia : Asymptotics

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(2017年11月25日) Bachmann-Landau symbols: Big-O (and relatives).
The most common symbol in a system of four asymptotic notations.

The big-O symbol was introduced by Paul Bachman in 1824. In 1909, Edmund Landau adopted that notation.

At the same time, Landau introduced the same syntax for an unrelated little-o notation which pertains more to pure theoretical asymptotic analysis. In fact, it just expresses negligibility in the above sense. Thus, as x tends to L, we have three equivalent notations:

f (x) << g (x) or f (x) = o (g x) or f (x) / g (x) ® 0

The third one is the defining relation for scalar quantities only (where division is defined) but the first two are well-defined for normed vector spaces as well, with the understanding that a vector function is negligible compared to another exactly when the norm of the first is negligible compared to the norm of the second.

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Paul Bachmann (1837-1920) | Edmund Landau (1877-1938) | Landau's symbol

Wikipedia : Big O notation

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(2016年01月14日) Solving Asymptotic Equations
The method of dominant balance.

If, for a given limit point L, we have:

f (x) ~ g (x) + h (x)
with h (x) << g (x)

Then, we have f (x) ~ g (x)

That makes asymptotic equivalences easier to solve than algebraic equations.

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Dominant balance and perturbations (StackExchange) | Method of dominant balance

Wikipedia : Method of dominant balance

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(2012年10月04日) Asymptotic expansions about a limit point.
Asymptotic expansions may or may not be convergent.

Against proper mathematical usage, the term asymptotic series is used exclusively for divergent series by several leading authors (including R.B. Dingle and Gradshteyn & Ryzhik ). I beg to differ.

It makes a lot more sense to work out an asymptotic expansion first and only then worry whether it converges or not (which is usually far from obvious. Likewise, asymptotic expansions are best defined without concerns about possible convergence:

Definition :

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Bob Dingle has investigated how the exact values of a function can be extracted from the latent information contained in its asymptotic expansion, even if it's not convergent.

Asymptotic Expansions: Their Derivation and Interpretation by R.B. Dingle (1973, 521 pp.).

Robert Balson Dingle (1926-2010; PhD 1952) by Sir Michael Berry & John Cornwell.

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(2012年08月03日) Stieltjes Functions & Moments

Thomas Stieltjes (1856-1894)
Thomas Stieltjes

Well before the more general notion of distributions was devised (in 1944, by my late teacher Laurent Schwartz) the Dutch mathematician Thomas Stieltjes considered measures as generalized derivatives of functions of bounded variations of a real variable. Such functions are differences of two monotonous bounded functions; they need not be differentiable or continuous. (Stieltjes got his doctorate in Paris, under Hermite and Darboux.)

Let's define a weight function r as a nonnegative function of a nonnegative variable which has a moment of order n, expressed by the following convergent integral, for any nonnegative integer n :

a_n = ò ^¥ r(t) t ⁿ dt

₀

To any such weight function is associated a Stieltjes function defined by:

f (x) = ò ^¥ r(t) dt

Vinculum

₀ 1 + x t

A Sieltjes function f has four properties:

• It's analytic on the cut plane (.e., outside the negative real axis).
• It tends to zero at infinity along any direction of the cut plane.
• Its asymptotic series about 0 in the cut plane is S (-a)_n zⁿ
• Its opposite is Herglotz (i.e., sign Img -f (z) = sign Img z).

Surprisingly, the converse is true (those 4 properties imply f is Stieltjes).

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Stieltjes series | Stieltjes moment problem | Thomas Stieltjes (1856-1894)

Stieltjes theory (1:30:42) by Carl M. Bender (PI, 2011)

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James Stirling (the Venetian) 1692-1770 (2012年08月01日) Stirling Approximation & Stirling Series
The Stirling series is a divergent asymptotic series.

In 1730, Abraham de Moivre (1667-1754) showed that:

Log n! = n Log n - n + ½ Log n + O(1)

His younger friend James Stirling (1692-1770) immediately refined that by finding that the actual limit of the last term is Log Ö2p. That's just enough to give a proper asymptotic equivalence for n! , namely:

Stirling's Formula

Vinculum

n! ~ Ö 2pn ( n/e ) ⁿ

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Stirling approximation | Lanczos approximation | Implementation of the Gamma function

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(2016年03月12日) Hyper-Asymptotics
Asymptotics beyond all orders.

All of the above is sometimes called Poincaré asymptotics.

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Asymptotic, Superasymptotic and Hyperasymptotic Series by John P. Boyd (2000年08月21日).

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(2019年07月27日) Stokes Phenomenon
Asymptotic approximation valid only over complex wedges.

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Getting a grip on the Stokes Phenomenon by Bruno Eijsvoogel (MS Thesis, May 2017).

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(2019年08月08日) WKB Method (LG, JWKB, WKBJ)
Liouville-Green (1837), Jeffreys (1923), Wentzel-Kramers-Brillouin (1926).

A linear differential equation of order n has this form, with e = 1 :

... / ...

The method is to determine an asymptotic series of the solution about e = 0.

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Joseph Liouville (1809-1882, X1825) | George Green (1793-1841)
WKB approximation (1923, 1926) | Sir Harold Jeffreys (1891-1989)
Gregor Wentzel (1898-1978) | Hans Kramers (1894-1952) | Léon Brillouin (1889-1969)

Learning Geophysical Dynamics from Jeffreys (3:10) by Freeman Dyson..