On finite double integrals involving a general multivariable polynomial
and generalized multivariable Gimel-function
Frédéric Ayant
Teacher in High School , France E-mail :[email protected]
ABSTRACT
In this paper we evaluate three general finite double integrals involving the product of algebraic and exponential functions, a general multivariable polynomials and generalized multivariable polynomials. Some new and interesting special cases of our main integrals have been considered briefly.
Keywords : multivariable Gimel-function, hypergeometric function, finite double integrals, multivariable polynomials.
2010 Mathematics Subject Classification. 33C99, 33C60, 44A20
1. Introduction and preliminaries.
We define a generalized transcendental function of several complex variables.
= (1.1)
with
On Finite Double Integrals Involving a General
Multivariable Polynomial and Generalized
. . . .
. . . .
(1.2)
and
(1.3)
1) stands for .
2)
and verify :
.
3) .
4)
5)
The contour is in the - plane and run from to where if is a real number with loop, if necessary to ensure that the poles of
, to
the right of the contour and the poles of
, lie to the left of the
contour . The condition for absolute convergence of multiple Mellin-Barnes type contour (1.1) can be obtained of the corresponding conditions for multivariable H-function given by as :
where
(1.4)
Following the lines of Braaksma ([2] p. 278), we may establish the the asymptotic expansion in the following convenient form :
,
, where :
and
Remark 1.
If and
, then the generalized multivariable Gimel-function reduces in the generalized multivariable Aleph- function ( extension of multivariable Aleph-function defined by Ayant [1]).
Remark 2.
If and
, then the generalized multivariable Gimel-function reduces in a generalized multivariable I-function (extension of multivariable I-function defined by Prathima et al. [7]).
Remark 3.
If and
, then the generalized multivariable Gimel-function reduces in generalized of multivariable I-function (extension of multivariable I-function defined by Prasad [6]).
(1.5)
(1.6)
(1.7)
(1.8)
(1.9)
(1.10)
(1.11)
(1.12)
The general class ofpolynomials introduced by Srivastava [9] has been further generalized by Srivastava and Garg [10] to a multivariable polynomial in the following manner [10] :
(1.13)
where are arbitrary positive integers and coefficients
are arbitrary constants, real or complex.
We shall note
2. Required results.
We shall require the following integral ([4], p. 450), ([3], p. 10), ([5], p. 71) and ([8], p. 254) for the evaluation of our main integrals :
Lemma 1.
(2.1)
provided
Lemma 2.
(2.2)
provided .
Lemma 3.
(2.3)
provided
Lemma 4.
(2.4)
provided
3. Main integrals.
We shall evaluate the following general and new integrals :
(3.1)
provided
and
where is defined by (1.4).
Proof
To establish the integral (2.1), we first use the series representation of the multivariable polynomial with the help of (1.13) and express the generalized multivariable Gimel-function as Mellin-Barnes multiple integrals contour with the the help of (1.1), interchanging the order of summation and integration which is justified under the conditions mentioned above, we have (say I )
(3.2)
Evaluating the and -integrals with the help of Lemmae 4 and 2 respectively and Interpreting the resulting expression with the help of (1.1), we obtain the desired theorem 1.
Theorem 2.
(3.3)
Provided
and
where is defined by (1.4).
(3.4)
provided
and
where is defined by (1.4).
The proofs of theorems 2 and 3 are similar to that theorem 1 with the difference that here we make use of (2.2) and (2.4), (2.3) and (2.4) respectively instead of (1.3) and (1.6).
4. Special cases.
Taking and replacing by therein,ollowing we arrive at the
following general integrals
Corollary 1.
(4.1)
Taking in the (3.1), and evaluate the -integral, we obtain
Corollary 2.
(4.2)
5. Conclusion.
Similar type of integrals would follow from (2.2) and (2.3). The integrals (3.1), (3.2) and (3.3) are also quite general in nature. By suitably specializing the arbitrary coefficients in the general class of polynomials and the parameters of the generalized multivariable gimel-function, a large number of integrals can be evaluated.
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