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 ABSTRACT
The most voluminous bibliography of the analytical methods for calculating of integrals is represented in the article [19]. It is shown there that the most effective and the simplest algorithm of analytical integration was made by O.I. Marichev [8, 9, 12]. Later it was realized in the reference-books [16-18, 20]. This algorithm allows us to calculate definite and indefinite integrals of the products of elementary and special functions of hypergeometric type. It embraces about 70 per cent of integrals which are included in the world reference-literature. It allows to calculate many other integrals too.
The present article contains short description of this algorithm and its realization in the REDUCE system during the process of creation of INTEGRATOR system. Only one general method of integration is known to be realized on the computers, i.e. criterion algorithm for calculating of indefinite integrals of elementary functions through elementary functions by themselves (the authors of it are M. Bronstein and other).
The idea of our algorithm is in the following. The initial integrals is transformed to contour integral from the ratio of products of gamma-functions by means of Mellin transform and parseval equality. The residue theorem is used for the calculating of the received integral which due to the strict rules results in sums of hypergeometric series. The value of integral itself and the integrand functions are the special cases of the well-known Meijer's G-function [4, 7, 8, 12, 14, 18].
Programming packet is realized in programming languages PASCAL and REDUCE. It also offers the opportunity of finding the values for some classical integral transforms (Laplace, Hankel, Fourier, Mellin and etc.). The REDUCE's part of packet contains the main properties of the well-known special functions, such as the Bessel and gamma-functions and kindred functions, Anger function, Weber function, Whittaker functions, generalized hypergeometric functions. Special place in the packet is occupied by Meijers's G-function for which the main properties such as finding the particular cases and representation by means of hypergeometric series are realized.
REFERENCES
Note: OCR errors may be found in this Reference List extracted from the full text article. ACM has opted to expose the complete List rather than only correct and linked references.
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1
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Adamchik V.S. The singular cases of the functions of hypergeometric type and their applications. Doctor dissert. Minsk, 1987, 107 p. (in Russian).
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2
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Adamchik V.S., Kolbig K.S. A definite integral of a product of two polylogarifhms, SIAM J. Math. Anal. 1988. Vol. 19, No 4, p. 926-938.
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3
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Adamchik V.S., Marichev O.I. Representations of functions of hypergeometric type in logarithmic cases. Vestsi Akad. Navuk BSSR. Set. F iz.-Mat. Navuk, 1983, No 5, p. 29-35 (in Russian).
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4
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Braaksma B.L.J. In memoriam C.S.Meijer. Nieuw Arch. Voor Wiskunde (3). 1975. Vol. 23, p. 95-104.
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5
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Kalla S.L., Marichev O.I. Behaviour of hypergeometric function F p q-~ (z) in the vicinity of unity. Rev. Teen. Fac. Ingr. Univ. Zulia. Maracaibo, 1984. Vol. 7, No 2, p. 1-8.
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6
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Lafferty E.L. MACSYMA. 1979, p. 465-481.
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7
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Luke Y.L. The Special Functions and Their Approximations. In 2 vls. N.Y., Acad. Press, 1969. Vol. 1, 349 p.
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8
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Marichev O.I. Calculus of Integrals of Higher Transcendental Functions (the theory and tables of formulas). Minsk, Nauka i Tekhnika, 1978, 310 p. (in Russian).
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9
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Marichev O.I. A method for calculating integrals of hypergeometric functions. Dokl. Akad. Nauk BSSR, 1981. Vol. 25, No 7, p. 590-593 (in Russian).
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10
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Marichev O.I. Calculation of integral transformations of hypergeometric functions. Generalized functions and their applications in mathematical physics (Moscow, 1980), Akad. Nauk SSSR, Vychisl. Tsentr. Moscow, 1981, p. 323-331.
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11
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Marichev O.I. Conditions for the reversibility of Mellin-Barnes integrals. Dokl. Akad. Nauk BSSR, 1982. Vol. 26, No 3, p. 205-208. (in Russian).
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12
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Marichev O.I. Handbook of Integral Transforms of Higher Transcendental Functions, theory and algorithmic tables. Chichester, Ellis Horwood Ltd., 1983, 336 p.
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13
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Marichev O.I. Asymptotic behavior of functions of hypergeometric type. Vestsi Akad. Navuk BSSR. Ser. F iz.-Mat. Navuk, 1983, No 4, p. 18- 25. (in Russian).
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14
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Marichev O.i. On the representation of Meijer's G-function in the vicinity of singular unity. Complex Analysis and Applications'81 (Varna, 1981), Bulgar. Acad. Sci., Sofia, 1984, p. 383-398.
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15
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Nguyen Thanh Hai. The functions of hypergeometric type of two variables and two-dimensional integral transforms. Doctor. dissert. Minsk, 1990, 140 p. (in Russian).
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16
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Prudnikov A.P., Brychkov Yu.A., Marichev O.I. Integrals and Series. Vol. 1: Elementary functions, Gordon and Breach Sci. Publ., N. Y., London, Tokyo, 1986, 798 p. (first ed. in Moscow, Nauka, 1981).
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17
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Prudnikov A.P., Brychkov Yu.A., Marichev O.I. Integrals and Series. Vol. 2' Special functions, Gordon and Breach Sci. Publ., N. Y., London, Tokyo, 1986, 750 p. (first ed. in Moscow, Nauka, 1983).
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18
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Prudnikov A.P., Brychkov Yu.A., Marichev O.I. Integrals and Series. Vol. 3' More special functions, Gordon and Breach Sci. Publ., N. Y., London, Tokyo, 1989, 800 p. (first ed. in Moscow, Nauka, 1986).
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19
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Prudnikov A.P., Brychkev Yu.A., Marichev O.I. Calculus of integrals and Mellin transform. Itogi Nauki i Tekhniki, Math. analysis. Akad. Nauk SSSR. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1989, Vol. 27, p. 3-146 (in Russian).
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20
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Prudnikov A.P., Brychkov Yu.A., Marichev O.I. Integrals and Series. Vol. 4' Laplace transforms, Gordon and Breach Sci. Publ., N. Y., London, Tokyo, to appear in 1992, near 800 p.
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