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 Uspekhi Mat. Nauk, 1976, Volume 31, Issue 2(188), Pages 135–202 (Mi umn3682)

Some problems in the analytic theory of Feynman integrals

V. A. Golubeva

Abstract: This article contains a survey of the research during the last decade on the analytic theory of Feynman integrals. We give a combinatorial definition of a Feynman integral, the explicit form of the simplest Feynman integrals, also the equations of their Landau varieties and a concise characterization of them. The main part of the article contains an investigation of the analytic and asymptotic properties of the Feynman integral of a single-loop diagram in the zero-spin theory of the interactions of particles: we give its expansion in a generalized hypergeometric series, the system of partial differential equations satisfied by it, and the ramification properties of the integral on a Landau variety. The problems solved for this integral allow us to pose a number of interesting problems for an arbitrary convergent Feynman integral.

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English version:
Russian Mathematical Surveys, 1976, 31:2, 139–207

Bibliographic databases:

UDC: 517.5
MSC: 81T18, 81Q30, 33C20, 35Q15, 55Q05, 35F05

Citation: V. A. Golubeva, “Some problems in the analytic theory of Feynman integrals”, Uspekhi Mat. Nauk, 31:2(188) (1976), 135–202; Russian Math. Surveys, 31:2 (1976), 139–207

Citation in format AMSBIB
\Bibitem{Gol76} \by V.~A.~Golubeva \paper Some problems in the analytic theory of Feynman integrals \jour Uspekhi Mat. Nauk \yr 1976 \vol 31 \issue 2(188) \pages 135--202 \mathnet{http://mi.mathnet.ru/umn3682} \zmath{https://zbmath.org/?q=an:0334.28008|0342.28005} \transl \jour Russian Math. Surveys \yr 1976 \vol 31 \issue 2 \pages 139--207 \crossref{https://doi.org/10.1070/RM1976v031n02ABEH001487} 

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This publication is cited in the following articles:
1. A. A. Bolibrukh, “Pfaffian systems of Fuchs type on a complex analytic manifold”, Math. USSR-Sb., 32:1 (1977), 98–108
2. V. A. Golubeva, “On systems with regular singularities, and their solutions”, Math. USSR-Izv., 27:1 (1986), 27–38
3. A. B. Antonevich, “Boundary value problems with strong nonlocalness for elliptic equations”, Math. USSR-Izv., 34:1 (1990), 1–21
4. A. A. Bolibrukh, “The Riemann–Hilbert problem”, Russian Math. Surveys, 45:2 (1990), 1–58
5. V. A. Golubeva, V. P. Leksin, “Algebraic Characterization of the Monodromy of Generalized Knizhnik–Zamolodchikov Equations of $B_n$ Type”, Proc. Steklov Inst. Math., 238 (2002), 115–133
6. Golubeva V.A., “On the Riemann–Hilbert correspondence for generalized Knizhnik–Zamolodchikov equations for different root systems”, Differential Equations and Quantum Groups - ANDREY A. BOLIBRUKH MEMORIAL VOLUME, Irma Lectures in Mathematics and Theoretical Physics, 9, 2007, 189–207
7. Mikhail Yu. Kalmykov, Bernd A. Kniehl, “Towards all-order Laurent expansion of generalised hypergeometric functions about rational values of parameters”, Nuclear Physics B, 809:3 (2009), 365
8. V. A. Golubeva, “On the Regge–Gelfand problem of construction of a Pfaff system of Fuchsian type with a given singular divisor”, Journal of Mathematical Sciences, 202:5 (2014), 653–666
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