By Semen B. Yakubovich, Yurii F. Luchko (auth.)
ISBN-10: 9401045232
ISBN-13: 9789401045230
ISBN-10: 9401111960
ISBN-13: 9789401111966
The goal of this ebook is to improve a brand new process which we known as the hyper geometric one to the speculation of varied imperative transforms, convolutions, and their purposes to options of integro-differential equations, operational calculus, and review of integrals. we are hoping that this straightforward technique, with the intention to be defined less than, permits scholars, put up graduates in arithmetic, physicists and technicians, and critical mathematicians and researchers to discover during this e-book new fascinating leads to the idea of indispensable transforms, specific capabilities, and convolutions. the assumption of this procedure are available in numerous papers of many authors, yet systematic dialogue and improvement is learned during this e-book for the 1st time. allow us to clarify in brief the fundamental issues of this process. because it is understood, within the thought of distinctive services and its purposes, the hypergeometric capabilities play the most position. along with recognized undemanding services, this type contains the Gauss's, Bessel's, Kummer's, services et c. regularly case, the hypergeometric services are outlined as a linear combos of the Mellin-Barnes integrals. those ques tions are generally mentioned in bankruptcy 1. furthermore, the Mellin-Barnes variety integrals could be understood as an inversion Mellin remodel from the quotient of goods of Euler's gamma-functions. hence we're ended in the final construc tions just like the Meijer's G-function and the Fox's H-function.
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Extra info for The Hypergeometric Approach to Integral Transforms and Convolutions
Example text
2). Prudnikov et al. (1986)). 58) 11 (:z:21 -5/4 ) XG l :3 4" -1/4, -1/4, -3/4 . 59) r [s+1/4,s+3/4] (~)-2·ds V211"3/2i" s + 5/4,3/4 - s 2 l = _ 2:Z: Si (:Z:). Prudnikov et al. (1986)) is an automorphism in the space M- 1 (L) and its inverse has the form f(:z:) = 1 0 00 (1 - cos(:z:y) :z:y . ) + sm(:z:y) g(y)dy. 60): k(:z:) 2:z:. (2:Z:) 1/2 11 (:z:21 5/4 ) G1 :3 4 1/4, 1/4, 3/4 . 64) is understood in the L 2-sense. 7) is valid. Such kernels k(:z:), k(:z:) E K;* are called general Fourier kernels.
N". ;It aj > - 1/2' , J -_ ni + 1, ... R(I3~i») > -1/2, j = 1, ... R(I3~i») < 1/2, j = mi + 1, ... 112) ay) E R, a~2ni = _a~i), j = In particular, if Pi = 2ni, qi = 2mi, mi =f ni, 1, ... ,ni; l3~i) E R, 13~2mi = -13~i), j = 1, ... 111) is unitary in the space L 2 (R). 5 Chapter 2 Multidimensional Watson transforms In this last section we demonstrate the generalization of the one-dimensional Watson transforms to multidimensional case due to Nguyen Thanh Hai et ai. (1992). Brychkov et al. (1992), introduced in Vu Kim Tuan and Nguyen Thanh Hai (1991).
71) are necessary ones. 73) are valid for any function f(x) from L 2 (R+). We take f(u) = 1, u :S x, f(u) = 0, u > x. Then [Kf](u) = k1 (xu)/u and analogously [Kf](u) = k1 (yu)/u. 71). (1_-~s) x-Ids = O. 7). 7 is proved. Yakubovich (1992». 71) is fulfilled. m'M,N-+00 l-N M f(:z:) (cos(:z:y) + sin(:z:y»g(y)dy. 90) which can be applied to obtain the new Watson transform pairs. 71) holds. = Proof. 93) I)]). 31 Transforms With Arbitrary Kernels We introduce the quantities k~l)(1 Z I) = k(1 z I) + k(-I z I), k~2)(1 z I) = k(1 z I) - k(-I z I).
The Hypergeometric Approach to Integral Transforms and Convolutions by Semen B. Yakubovich, Yurii F. Luchko (auth.)
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