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References
H. Mellin, “Om definita integraler, hvilka för obegränsadt växende vörden af vissa heltaliga parametrar hafva till gränser hypergeometriska funktionen of särskilda ordningen,” Acta Soc. Sci. Fenn.,20, No. 7, 3–39 (1985).
E. W. Barnes, “The asymptotic expansion of integral functions defined by generalized hypergeometric series,” Proc. London Math. Soc.,5, No. 2, 59–116 (1907).
E. W. Barnes, “A new development of linear differential equations,” Trans. Cambr. Philos. Soc.,22, 178–221 (1908).
E. W. Barnes, “On asymptotic expansions of the integral functions\(\sum {_{n = 0}^\infty \frac{{\Gamma (1 + \alpha n)x^n }}{{\Gamma (1 + n)}}} \) and\(\sum {_{n = 0}^\infty \frac{{\Gamma (1 + n\theta )}}{{\Gamma (1 + n + n\theta )}}} \),” Trans. Cambr. Philos. Soc.,20, 215–232 (1906).
I. M. Gel’fand, M. I. Graev, and V. S. Retakh, “General hypergeometric systems of equations and series of hypergeometric type,” Uspekhi Mat. Nauk,47, No. 4, 3–82 (1992).
I. M. Gel’fand, A. V. Zelevinskiî, and M. M. Kapranov, “Hypergeometric functions and toric varieties,” Funktsional. Anal. i Prilozhen,23, No. 2, 12–26 (1989).
I. M. Gel’fand, M. I. Graev, and V. S. Retakh, “Theq-hypergeometric Gauss equation and the description of its solutions in the form of series and integrals,” Dokl. Akad. Nauk,331, No. 2, 140–143 (1993).
I. Gelfand, M. Kapranov, and A. V. Zelevinsky, “Generalized Euler integrals andA-hypergeometric functions,” Adv. Math.,84, No. 2, 255–271 (1990).
V. V. Batyrev, “Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori,” Duke Math. J.,69, No. 2, 349–409 (1993).
M. Passare, A. Tsikh, and O. Zhdanov, “A multidimensional Jordan residue lemma with an application to Mellin-Barnes integrals,” Aspects Math., 233–241 (1994).
F. Griffiths and J. Harris, Principles of Algebraic Geometry. Vol. 1 and 2 [Russian translation], Mir, Moscow (1982).
A. K. Tsikh, Multidimensional Residues and Their Applications [in Russian], Nauka, Novosibirsk (1988).
L. A. Aîzenberg and A. P. Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis [in Russian], Nauka, Novosibirsk (1979).
A. K. Tsikh, Methods of the Multidimensional Residue Theory [in Russian], Dis. Dokt. Fiz.-Mat. Nauk, Novosibirsk (1990).
Yu. V. Sidorov, M. V. Fedoryuk, and M. I. Shabunin, Lectures in the Theory of Functions of a Complex Variable [in Russian], Nauka, Moscow (1982).
H. Bateman and A. Erdélyi, Higher Transcendental Functions. Vol. 1–3 [Russian translation], Nauka, Moscow (1966).
M. Abramowitz and I. A. Stegun, eds, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables [Russian translation], Nauka, Moscow (1979).
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The research was financially supported by the Krasnoyarsk Regional Science Foundation (Grant 4F0228) and the Russian Foundation for Basic Research (Grant 96-01-00080).
Krasnoyarsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 39, No. 2, pp. 281–298, March–April, 1998.
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Zhdanov, O.N., Tsikh, A.K. Studying the multiple Mellin-Barnes integrals by means of multidimensional residues. Sib Math J 39, 245–260 (1998). https://doi.org/10.1007/BF02677509
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DOI: https://doi.org/10.1007/BF02677509