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© 2000-2017   Gérard P. Michon, Ph.D.

Asymptotic Analysis

The scientist knows very well that he is approaching ultimate truth only in an asymptotic curve and is barred from ever reaching it.
  Konrad Lorenz (1903-1989) in  "On Agression" (1963)

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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  (2005-12-12)
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 (2006-10-11)
Euler characteristic of a category as the sum of a divergent series (2007-07-09)
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  (1995-03-15).
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)

Asymptotic Methods in Analysis  by  N.J. de Bruijn  (1958, Dover 1981).
Asymptotic Expansions: Derivation and Interpretation  R.B. Dingle  (1973).
Asymptotic Methods and Perturbation Theory.   Bender  &  Orszag  (1978).

Videos :   Mathematical Physics  by  Carl M. Bender  (PSI, 2010)
 Carl Bender at the blackboard
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15

The Art of Asymptotic Approximation (53:06)  Frank Leppington  (LMS, 1989).
Asymptotics (58:58)  Don Zagier  (2014-11-06).


Asymptotic Analysis

(2012-08-01)   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

Note that this relation is  unrelated  to ordering in the real line.  For example, both of the following relations are true as  x  tends to zero:

-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
As  x  tends to  L,  one vectorial quantity  f (x)  is said to be  negligible  compared to another quantity  g (x)  when the ratio   || f (x) || / || g (x) ||   has a limit of zero.
Two quantities are  asymptotically equivalent to  each other  ("asymptotic to"  or  "equivalent to", for short)  if their difference is negligible compared to their sum.

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

(2017-11-25)   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.

 Come back later, we're
 still working on this one...

Paul Bachmann (1837-1920)   |   Edmund Landau (1877-1938)   |   Landau's symbol
Wikipedia :   Big O notation

(2016-01-14)   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.

 Come back later, we're
 still working on this one...

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

(2012-10-04)   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  concern about possible convergence:

Definition :

 Come back later, we're
 still working on this one...

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.

(2012-08-03)   Stieltjes Functions  &  Moments
   Thomas Stieltjes 
Thomas Stieljes

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 :

an   =    ò  ¥  r(t)  t n  dt

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

f (x)   =    ò  ¥    r(t)  dt
0 1 + x t

 Come back later, we're
 still working on this one...

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

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

 Come back later, we're
 still working on this one...

Stirling approximation   |   Lanczos approximation   |   Implementation of the Gamma function

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

All of the above is sometimes called  Poincaré asymptotics.

 Come back later, we're
 still working on this one...

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