## Feynman Integral CalculusThis 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. |

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### Contents

III | 11 |

V | 14 |

VI | 19 |

VII | 21 |

VIII | 25 |

IX | 31 |

XI | 34 |

XII | 35 |

XLI | 171 |

XLII | 173 |

XLIV | 178 |

XLV | 182 |

XLVI | 184 |

XLVII | 185 |

XLVIII | 191 |

XLIX | 194 |

XIII | 36 |

XIV | 38 |

XV | 43 |

XVI | 45 |

XVII | 54 |

XVIII | 56 |

XIX | 58 |

XX | 65 |

XXI | 68 |

XXII | 74 |

XXIII | 84 |

XXIV | 95 |

XXV | 102 |

XXVI | 105 |

XXVII | 109 |

XXVIII | 112 |

XXIX | 115 |

XXX | 116 |

XXXI | 121 |

XXXII | 128 |

XXXIII | 135 |

XXXIV | 138 |

XXXV | 139 |

XXXVII | 144 |

XXXVIII | 153 |

XXXIX | 159 |

XL | 169 |

### Common terms and phrases

algorithm alpha parameters alpha representation analytic continuation Appendix apply asymptotic asymptotic expansions auxiliary basic polynomial Buchberger algorithm calculations Chap coeﬃcient function consider constructed convergence ddkddl deﬁned deﬁnition denominator denote derive diagram of Fig diﬀerent diﬀerential equations dimensionally regularized example expansion external momenta factor Feynman diagrams Feynman integrals Feynman parameters ﬁnite ﬁrst formulae gamma functions given integral Gr¨obner basis graph IBP relations indices integrand integration contour irreducible J_ioo Laurent series Lett linear combination lines ln(l loop momenta Mandelstam variable masses massive massless on-shell master integrals MB integrals MB representation method momentum monomials non-positive non-trivial Nucl numerator obtain the following on-shell on-shell box one-loop parametric integrals Phys pole polylogarithms powers procedure propagators r(Ai recurrence relations reduction problem right-hand side scalar Sect sector singularities solve the reduction strategy subgraphs taking residues terms of gamma two-loop values variables z)+H zero