Feynman Integral Calculus

The goal of the book is to summarize those methods for evaluating Feynman integrals that have been developed over a span of more than fifty years. The book characterizes the most powerful methods and illustrates them with numerous examples.

This is a textbook version of my previous book [190]. Problems and solutions have been included, Appendix G has been added, more details have been presented, recent publications on evaluating Feynman integrals have been taken into account and the bibliography has been updated. 1 ThegoalofthebookistodescribeindetailhowFeynmanintegrals canbe evaluatedanalytically. TheproblemofevaluatingLorentz-covariantFeynman integrals over loop momenta originated in the early days of perturbative quantum ? eld theory. Over a span of more than ? fty years, a great variety of methodsforevaluatingFeynmanintegralshasbeendeveloped. Mostpowerful modern methods are described in this book. Iunderstandthatifanotherperson-inparticularoneactivelyinvolvedin developing methods for Feynman integral evaluation - wrote a book on this subject, he or she would probably concentrate on some other methods and would rank the methods as most important and less important in a di? erent order. I believe, however, that my choice is reasonable. At least I have tried to concentrate on the methods that have been used recently in the most sophisticated calculations, in which world records in the Feynman integral 'sport' were achieved.

Feynman Integrals: Basic Definitions and Tools. - Evaluating by Alpha and Feynman Parameters. - Evaluating by MB Representation. - IBP and Reduction to Master Integrals. - Reduction to Master Integrals by Baikov s Method. - Evaluation by Differential Equations. - Tables. - Some Special Functions. - Summation Formulae. - Table of MB Integrals. - Analysis of Convergence and Sector Decompositions. - A Brief Review of Some Other Methods. - Applying Gröbner Bases to Solve IBP Relations. - Solutions.