### Abstract

This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. The unifying concept for convergence acceleration methods is that in many cases, one can reach much faster convergence than by adding a particular series term by term. In some cases, it is even possible to use a divergent input series, together with a suitable sequence transformation, for the construction of numerical methods that can be applied to the calculation of special functions. This review both aims to provide some practical guidance as well as a groundwork for the study of specialized literature. As a second topic, we review some recent developments in the field of Borel resummation, which is generally recognized as one of the most versatile methods for the summation of factorially divergent (perturbation) series. Here, the focus is on algorithms which make optimal use of all information contained in a finite set of perturbative coefficients. The unifying concept for the various aspects of the Borel method investigated here is given by the singularities of the Borel transform, which introduce ambiguities from a mathematical point of view and lead to different possible physical interpretations. The two most important cases are: (i) the residues at the singularities correspond to the decay width of a resonance; and (ii) the presence of the singularities indicates the existence of nonperturbative contributions which cannot be accounted for on the basis of a Borel resummation and require generalizations toward resurgent expansions. Both of these cases are illustrated by examples.

Original language | English |
---|---|

Pages (from-to) | 1-96 |

Number of pages | 96 |

Journal | Physics Reports |

Volume | 446 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - Jul 2007 |

### Fingerprint

### Keywords

- Asymptotic problems and properties
- Computational techniques
- Numerical approximation and analysis
- Quantum electrodynamics (specific calculations)

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Physics Reports*,

*446*(1-3), 1-96. https://doi.org/10.1016/j.physrep.2007.03.003

**From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions.** / Caliceti, E.; Meyer-Hermann, M.; Ribeca, P.; Surzhykov, A.; Jentschura, U.

Research output: Contribution to journal › Article

*Physics Reports*, vol. 446, no. 1-3, pp. 1-96. https://doi.org/10.1016/j.physrep.2007.03.003

}

TY - JOUR

T1 - From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

AU - Caliceti, E.

AU - Meyer-Hermann, M.

AU - Ribeca, P.

AU - Surzhykov, A.

AU - Jentschura, U.

PY - 2007/7

Y1 - 2007/7

N2 - This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. The unifying concept for convergence acceleration methods is that in many cases, one can reach much faster convergence than by adding a particular series term by term. In some cases, it is even possible to use a divergent input series, together with a suitable sequence transformation, for the construction of numerical methods that can be applied to the calculation of special functions. This review both aims to provide some practical guidance as well as a groundwork for the study of specialized literature. As a second topic, we review some recent developments in the field of Borel resummation, which is generally recognized as one of the most versatile methods for the summation of factorially divergent (perturbation) series. Here, the focus is on algorithms which make optimal use of all information contained in a finite set of perturbative coefficients. The unifying concept for the various aspects of the Borel method investigated here is given by the singularities of the Borel transform, which introduce ambiguities from a mathematical point of view and lead to different possible physical interpretations. The two most important cases are: (i) the residues at the singularities correspond to the decay width of a resonance; and (ii) the presence of the singularities indicates the existence of nonperturbative contributions which cannot be accounted for on the basis of a Borel resummation and require generalizations toward resurgent expansions. Both of these cases are illustrated by examples.

AB - This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. The unifying concept for convergence acceleration methods is that in many cases, one can reach much faster convergence than by adding a particular series term by term. In some cases, it is even possible to use a divergent input series, together with a suitable sequence transformation, for the construction of numerical methods that can be applied to the calculation of special functions. This review both aims to provide some practical guidance as well as a groundwork for the study of specialized literature. As a second topic, we review some recent developments in the field of Borel resummation, which is generally recognized as one of the most versatile methods for the summation of factorially divergent (perturbation) series. Here, the focus is on algorithms which make optimal use of all information contained in a finite set of perturbative coefficients. The unifying concept for the various aspects of the Borel method investigated here is given by the singularities of the Borel transform, which introduce ambiguities from a mathematical point of view and lead to different possible physical interpretations. The two most important cases are: (i) the residues at the singularities correspond to the decay width of a resonance; and (ii) the presence of the singularities indicates the existence of nonperturbative contributions which cannot be accounted for on the basis of a Borel resummation and require generalizations toward resurgent expansions. Both of these cases are illustrated by examples.

KW - Asymptotic problems and properties

KW - Computational techniques

KW - Numerical approximation and analysis

KW - Quantum electrodynamics (specific calculations)

UR - http://www.scopus.com/inward/record.url?scp=34250197550&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34250197550&partnerID=8YFLogxK

U2 - 10.1016/j.physrep.2007.03.003

DO - 10.1016/j.physrep.2007.03.003

M3 - Article

AN - SCOPUS:34250197550

VL - 446

SP - 1

EP - 96

JO - Physics Reports

JF - Physics Reports

SN - 0370-1573

IS - 1-3

ER -