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.

arXiv commentLatex file, 61 pages

References in corpus (37)

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• The critical exponents of the superfluid transition in He4
• The transition temperature of the dilute interacting Bose gas
• Transition temperature of a dilute homogeneous imperfect Bose gas
• Pushing the Limits of Monte Carlo Simulations for the 3d Ising Model
• The transition temperature of the dilute interacting Bose gas for $N$ internal degrees of freedom
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• Improved Convergence Proof of the Delta Expansion and Order Dependent Mappings
• Self-Similar Factor Approximants
• Monte Carlo simulation of O(2) phi^4 field theory in three dimensions
• Summation of Power Series by Self-Similar Factor Approximants
• Self-Similar Exponential Approximants
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• Temporal Dynamics in Perturbation Theory
• Self-Similar Structures and Fractal Transforms in Approximation Theory
• Self-Similar Interpolation in Quantum Mechanics
• Critical Indices as Limits of Control Functions
• Self-Similar Bootstrap of Divergent Series
• Algebraic Self-Similar Renormalization in Theory of Critical Phenomena
• Monte Carlo studies of three-dimensional O(1) and O(4) \boldmath{${\varphi }^4$} theory related to BEC phase transition temperatures
• Non-universal Critical Quantities from Variational Perturbation Theory and Their Application to the BEC Temperature Shift
• Statistical Green's Functions
• Extreme events in population dynamics with functional carrying capacity
• Method of self-similar factor approximants
• Extrapolation of power series by self-similar factor and root approximants
• Bose-Einstein condensation temperature of weakly interacting atoms
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• Self-similar continued root approximants
• Theory of Perturbations with a Strong Interaction
• Interplay Between Approximation Theory and Renormalization Group
• Statistical systems with nonintegrable interaction potentials
• Self-similar extrapolation in quantum field theory
• Critical Temperature in Weakly Interacting Multicomponent Field Theory
• Improved Taylor Expansion method in the Ising model
• Spectrum of Coherent Modes for Trapped Bose Gas