International Research Journal of Engineering and Technology (IRJET) Volume: 04 Issue: 05 | May -2017
www.irjet.net
e-ISSN: 2395 -0056 p-ISSN: 2395-0072
FOURIER SERIES INVOLVING H–FUNCTION OF TWO VARIABLES Mehphooj Beg1, Dr. S. S. Shrivastava2 VITS, Satna (M. P.), Institute for Excellence in Higher Education Bhopal (M. P.) --------------------------------------------------------------***-------------------------------------------------------------Abstract The object of this paper is to establish some new Fourier series involving H–function of two variables. 1. Introduction: Recently Mittal and Gupta [1, p. 117] has given the following notation of the H-function of two variables as:
=
(
)
(
)
(
)
(
)
(1)
∫ ∫
where ∏ ∏
∏
∏
∏
∏
∏
∏
∏
∏
∏
x and y are not equal to zero, and an empty product is interpreted as unity pi, qi, ni and mj are non negative integers such that pi ≥ ni ≥ 0, qi ≥ 0, qj ≥ mj ≥ 0, (i = 1, 2, 3; j = 2, 3). Also, all the A’s, α’s, B’s, β’s, γ’s, ’s, E’s, and F’s are assumed to the positive quantities for standardization purpose. The contour L1 is in the -plane and runs from – i∞ to + i∞, with loops, if necessary, to ensure that the poles of (dj j) (j = 1, ..., m2) lie to the right, and the poles of (1 – cj + γj) (j = 1, ..., n2), (1 – aj+ αj+ Aj) (j = 1, ..., n1) to the left of the contour. The contour L2 is in the -plane and runs from – i∞ to + i∞, with loops, if necessary, to ensure that the poles of (fj – Fj) (j = 1, ..., m3) lie to the right, and the poles of (1 – ej + Ej) (j = 1, ..., n3), (1 – aj+ αj + Aj) (j = 1, ..., n1) to the left of the contour. The function, defined by (1), is analytic function of x and y if ∑ ∑ ∑ R=∑ ∑ ∑ ∑ R=∑ The H-function of two variables given by (1) is convergent if ∑ ∑ ∑ ∑ ∑ U= ∑
(2)
V=
(3)
∑
∑
∑
∑
∑
∑
and
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