By Vladimir A. Smirnov
Introduction.- Feynman Integrals: easy Definitions and Tools.-Evaluating through Alpha and Feynman Parameters.- quarter Decompositions.- comparing through Mellin-Barnes Representation.- Integration via elements and relief to grasp Integrals.- review by means of Differential Equations.- comparing grasp Integrals by means of Dimensional Recurrence and Analyticity.- Asymptotic Expansions in Momenta and Masses.- Tables.- a few designated capabilities- Summation Formulae.- desk of MB Integrals.- a quick overview of a few different tools
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Extra info for Analytic tools for Feynman integrals
5. 47). 5 Properties of Dimensionally Regularized Feynman Integrals 29 Fig. 5 Tadpole Observe that one can trace the derivation of the integrals tabulated in Sect. 1 and see that the integrals are convergent in some non-empty domains of the complex parameters λl and ε and that the results are analytic functions of these parameters with UV, IR and collinear poles. Before continuing our discussion of setting scaleless integrals to zero, let us present an analytic result for the one-loop massless triangle integral with two on-shell external momenta, p12 = p22 = 0.
Here the limiting value of the regularization parameter is M = ∞. If we replace the integer powers al in the propagators by general complex numbers λl we obtain an analytically regularized  Feynman integral where the divergences of the diagram are encoded in the poles of this regularized quantity with respect to the analytic regularization parameters λl . 9) leads to the ∞ divergent behaviour Λ dr r λ1 +λ2 −3 , which results in a pole 1/(λ1 + λ2 − 2) at the limiting values of the regularization parameters λl = 1.
37), introduce new variables by αl = ηαl for all l = 1, 2, . . 39). 38) corresponding to Feynman diagrams with standard propagators but also for the alpha representation derived for Feynman diagrams with various linear propagators. 38) can greatly simplify the evaluation. 38)—see Sect. 1 of  and references therein. In addition to alpha parameters, the closely related Feynman parameters are often used. For a product of two propagators, one writes down the following relation: 1 (m 21 − p12 )λ1 (m 22 − p22 )λ2 = Γ (λ1 + λ2 ) Γ (λ1 )Γ (λ2 ) dξ ξ λ1 −1 (1 − ξ)λ2 −1 1 (m 21 − p12 )ξ + (m 22 − p22 )(1 − ξ) 0 λ1 +λ2 .
Analytic tools for Feynman integrals by Vladimir A. Smirnov