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On Meijer transform III

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On Meijer transform III COMPOSITIO MATHEMATICA J. P. JAISWAL OnMeijer transform III Compositio Mathematica, tome 12 (1954-1956), p. 284-297. <http://www.numdam.org/item?id=CM_1954-1956__12__284_0> © Foundation Compositio Mathematica, 1954-1956, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ On Meijer Transform III 1) by J. P. Jaiswal 1. Meijer [1] introduced the intégral equation where Wk,m(z) is Whittaker’s confluent hypergeometric function. In this equation f(t) is known as the original of ~(s), p(.9) the image of f(t) and (A) is symbolically denoted by [2] Particular cases of Meijer transform are :- (a) when k = 20131 2, (A) reduces to a) and will be known as Km-transform, and symbolically denoted as and will be known as D,,-transform, and symbolically denoted as (c) when k = ::l: m, (A) reduces to the Laplace Integral, 1) This paper is in continuation of my earlier papers [2] and [3]. 2) K.(z) and Dn(z) are the Bessel function and the parabolic cylinder function respectively. 285 which will be denoted symbolically as In this paper we have established some more properties of the Meijer transform by utilising integral representations of the Whittaker’s function, the parabolic cylinder function, and the Bessel function, and also the integrals involving these functions. The analogues of these properties are not known in the case of the Laplace transform. The conditions imposed on the theorems may in some cases be relaxed by analytic continuation. We have illus- trated this fac

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