From Asymptotic Series to Self-Similar Approximants

Abstract

The review presents the development of an approach of constructing approximate solutions to complicated physics problems, starting from asymptotic series, through optimized perturbation theory, to self-similar approximation theory. The close interrelation of underlying ideas of these theories is emphasized. Applications of the developed approach are illustrated by typical examples demonstrating that it combines simplicity with good accuracy.

Methods of Retrieving Large-Variable Exponents

“…Equations ( 15) uniquely define all parameters A j and n j for the even orders k of expansion (3). For the odd orders k, an additional normalization condition is required which, based on scaling arguments, implies that one of the A j can be set to one [14,15]. The other possibility could be by optimizing the factor approximant (12) with respect to one of A j .…”

Casimir interactions from infinite range and dilation symmetry

“…where {a i } are constants, x is considered to be an integer and x > 1. One simple way to realise the sum of this divergent series (S = ∞ i=0 a i x i ) in case of a i+1 /a i ≥ 1 is by analytic continuation using self-similar function [29,30,31,32] such as continued fraction [33,34,35]…”

Calculating critical temperature and critical exponents by self-similar approximants

“…In the vicinity of a fixed point, the self-similar relation holds [10,11]. The family {y k (f )}, with relation (18), composes an approximation cascade.…”

“…Employing the ideas described above, a convenient type of approximants has been derived, called self-similar factor approximants [10,11,15,16]. Let an asymptotic expansion for a real function be given:…”

“…In the vicinity of a fixed point, the evolution equation acquires the form of a self-similar relation, which justifies the name of the self-similar approximation theory. Mathematical details can be found in the review articles [10,11]. Here we only give the main ideas of the general theory in Sec.…”