By Vladimir A. Smirnov

Introduction.- Feynman Integrals: easy Definitions and Tools.-Evaluating by way of Alpha and Feynman Parameters.- zone Decompositions.- comparing through Mellin-Barnes Representation.- Integration through components and relief to grasp Integrals.- review by means of Differential Equations.- comparing grasp Integrals via Dimensional Recurrence and Analyticity.- Asymptotic Expansions in Momenta and Masses.- Tables.- a few distinct features- Summation Formulae.- desk of MB Integrals.- a quick overview of a few different equipment

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A. Smirnov, Nucl. Phys. B 554, 391 (1999) A. ), Higher Transcendental Functions, vols. M. E. Shilov, Generalized Functions, vol. G. Grozin, Heavy Quark Effective Theory (Springer, Heidelberg, 2004) K. Hepp, Commun. Math. Phys. 2, 301 (1966) G. ’t Hooft, M. Veltman, Nucl. Phys. B 44, 189 (1972). G. Leibbrandt, Rev. Mod. Phys. P. , Phys. Rev. V. B. Wise, Heavy Quark Physics (Cambridge University Press, Cambridge, 2000) N. L. van Neerven, Nucl. Phys. B 268, 453 (1986) M. Neubert, Phys. Rep. 245, 259 (1994) W.

Higher Transcendental Functions, vols. M. E. Shilov, Generalized Functions, vol. G. Grozin, Heavy Quark Effective Theory (Springer, Heidelberg, 2004) K. Hepp, Commun. Math. Phys. 2, 301 (1966) G. ’t Hooft, M. Veltman, Nucl. Phys. B 44, 189 (1972). G. Leibbrandt, Rev. Mod. Phys. P. , Phys. Rev. V. B. Wise, Heavy Quark Physics (Cambridge University Press, Cambridge, 2000) N. L. van Neerven, Nucl. Phys. B 268, 453 (1986) M. Neubert, Phys. Rep. 245, 259 (1994) W. Pauli, F. Villars, Rev. Mod. Phys.

K 2 )λ1 [−(q − k)2 ]λ2 (−2v·k − i0)λ3 Here v · q = 0. ) One of the propagators is not quadratic so that we proceed by alpha parameters and represent each of the three factors as an alpha integral. After taking a Gaussian integral over k we obtain iλ1 +λ2 +λ3 +ε−1 π d/2 l Γ (λl ) × exp i q 2 α1 α2 ∞ 0 ∞ 0 ∞ 0 − v 2 α32 α1 + α2 3 αlλl −1 dαl (α1 + α2 )ε−2 l=1 . 3). 27). 42) for the formal Fourier transformation within dimensional regularization. 6). In fact, the initial integral is nothing but the convolution of the two functions, f˜i = 1/(−k 2 − i0)λi , i = 1, 2.