Developments of the I-Function Several Complex Variables

Functions defined on the complex coordinate space are the subject of the mathematical field known as the theory of functions of many complex variables. , meaning complex numbers in n-tuple form. The Mathematics Subject Classification includes a top-level heading titled "several complex variables (and analytic space)" for the discipline that studies the characteristics of these functions. The functions under study are holomorphic or complex analytic, such that, locally, they are power series in the variables zi, just as they are in the complex analysis of functions of one variable, which is the case n = 1. They may also be thought of as locally square-integrable solutions to the n-dimensional Cauchy-Riemann equations or as locally uniform limits of polynomials. Any given domain for a complex variable, is holomorphic to some function, or, put another way, every domain is holomorphic to some function. However, this is not the case when there are several complicated factors involved. It is not necessarily the realm of holomorphy, hence the concept of holomorphy's domain is one of the major topics in this area of study. The Cousin issue is the challenge of constructing a global meromorphic function from zeros and poles, or "patching" the local data of meromorphic functions. Furthermore, the study of compact complex manifolds and complex projective varieties relies heavily on the intriguing phenomena that occur in many complex variables. has a unique flavor compared to other forms of complex analytic geometry, , or on Stein manifolds, are more akin to complex analytic geometry than the study of algebraic varieties.

Rezepin, Alexandr et.al (2017). This article discusses the production function model as a tool for describing the production process and finding solutions to real-world issues like determining the most appropriate production technology or allocating capital efficiently. The manufacturing process is analyzed using Urals Stampings Plant as a case study. There are two elements to the analysis. In the first section, we utilize regression analysis to assess the production function and determine the revenue elasticity of the primary resources. Using artificial neural networks, the authors of the second section of the investigation look into how the dynamics of the number of production variables and productivity affect the issue's physical volume. In conclusion, the authors provide suggestions for improving the efficiency of the Urals Stampings Plant's investment plan. Gerck, Edgardo et.al (2019). This paper introduces new avenues for the use of complex analysis in the physical and engineering sciences. Despite the fact that there is no such thing as a "complex number," it does provide a summary of the themes explored throughout four separate studies. The origin and destination of results are in real number theory; however, the complex plane plays a role and is not

functions; subharmonic analysis; and the residue theorem. Shadab, Mohd et.al (2018). In this paper, we introduce an extended beta function via the Mittag-Leffler function and explore various properties and formulas of the Mohd Shadab, Saima Jabee, and Junesang Choi 236 extended beta function, including integral repr, as a result of the widespread use of Euler's beta, hypergeometric, and confluent hypergeometric functions and their extensions in a wide variety of research fields, including engineering, chemical, and physical problems. We also give a graphical representation of the expanded beta function and a tabular summary of its specialized values, both of which may be calculated with relative ease by means of object-oriented programming languages. Rachel Webb, (2023) For I-functions of quasimaps, we establish their abelian-nonabelian correspondence. If Z is an affine l.c.i. variety under the action of a complex reductive group G, then we provide a rigorous proof of a formula that relates the I-functions of the quasimaps of the geometric invariant theory quotients Z//G and Z//T, where T is a maximal torus of G. We use the formula to determine the J-functions of certain Grassmannian bundles on Calabi-Yau hypersurfaces in Grassmannian varieties. Guala, Francesco et.al (2019). Define an institution. And what makes one institution different from another? Token institutions are concrete solutions to these difficulties, or strategic game equilibria, and are the focus of our answers to these concerns from a functionalist perspective. The functionalist perspective sheds light on how far institutions like marriage, property, and democracy may be altered before they cease to be the same thing altogether.

The following defines and illustrates the generalized Fox H-function, often known as the I-function of "r" variables: where 𝑖 = 1, . . ., 𝑟. Also, (i) 𝑧𝑖 = 0 , for 𝑖 = 1, . . ., 𝑟; (ii) 𝑖 = √−1; (iii) An empty product is equivalent to oneness; (iv) the parameters 𝑚𝑗, 𝑛𝑗, 𝑝𝑗, 𝑞𝑗 (𝑗 = 1, . . ., 𝑟), 𝑛, 𝑝, there exists a pair of positive integers and q such that 0≤𝑛≤𝑝, 𝑞≥0, 0≤𝑛𝑗 ≤ 𝑝𝑗, and 0≤𝑚𝑗 ≤ 𝑞𝑗 (𝑗 = 1, . . ., 𝑟) (not all zero simultaneously); (v) 𝛼(𝑖) 𝑗 (𝑗 = 1, . . ., 𝑝, 𝑖 = 1, . . ., 𝑟), 𝛽(𝑖) 𝑗 (𝑗 = 1, . . ., 𝑞, 𝑖 = 1, . . ., 𝑟), 𝛾(𝑖) 𝑗 (𝑗 = 1, . . ., 𝑝𝑖, 𝑖 = 1, . . ., 𝑟), and 𝛿(𝑖) 𝑗 (𝑗 = 1, . . ., 𝑞𝑖, 𝑖 = 1, . . ., 𝑟) are standardized to be positive numbers for convenience. Even if some of the values are 0 or negative, the definition of the I-function of "r" variables will still make sense. In a subsequent section, we will provide possible transformation formulae for these; (vi) 𝑎𝑗 (𝑗 = 1, . . ., 𝑝), 𝑏𝑗 (𝑗 = 1, . . ., 𝑞), 𝑐 (𝑖) 𝑗 (𝑗 = 1, . . ., 𝑝𝑖, 𝑖 = 1, . . ., 𝑟), and 𝑑(𝑖) 𝑗 (𝑗 = 1, . . ., 𝑞𝑖, 𝑖 = 1, . . ., 𝑟) are complex numbers; (vii) the exponents 𝐴𝑗 (𝑗 = 1, . . ., 𝑝), 𝐵𝑗 (𝑗 = 1, . . ., 𝑞), 𝐶(𝑖) 𝑗 (𝑗 = 1, . . ., 𝑝𝑖, 𝑖 = 1, . . ., 𝑟), and 𝐷(𝑖) 𝑗 (𝑗 = 1, . . ., 𝑞𝑖, 𝑖 = 1, . . ., 𝑟) values that are not integers for the different gamma functions used in (viii) The Mellin Barnes type Li contour in the complicated si-plane extends from 𝑐 −𝑖∞ to 𝑐 +𝑖∞, (𝑐 singularities of Γ𝐷(𝑖) 𝑗 (𝑑(𝑖) 𝑗 − 𝛿(𝑖) 𝑗 𝑠𝑖), 𝑗 = 1, . . ., 𝑚𝑖 lie to the right and Γ𝐶(𝑖) 𝑗 (1 − 𝑐(𝑖) 𝑗 + 𝛾(𝑖) 𝑗 𝑠𝑖), 𝑗 = 1, . . ., 𝑛𝑖 and Li's left side. Braaksma stablished that the I-function of "r" variables is analytic if and only if.

Integrals involving the product of the I-function, the Bessel function, and the General class of polynomials have been obtained in this subsection.

Proof: Let The I-function of one variable is expressed in terms of a Mellin-Barne type contour integral, which is obtained by generalizing the class of polynomials and Bessel's function in series form.

order of integration.

Using x-integral evaluation with equation, we get

by shuffling the words around, we obtain

The right-hand side of equation is found by expressing the contour integral of Mellin-Barne's type in terms of an I-function of one variable.

Here, we obtain three integrals for the combination of the I-function in two variables provided by Goyal and Agrawal and the second class of multivariable polynomials given by Srivastava.

The necessary conditions for equation to hold are (i)

(iii)

(i) (ii) (iii) Explanation of Proof: Let By writing the series expansion of the polynomials and the I-function in terms of a double Mellin-Barne's type contour integral, we get If we are allowed to alter the order of integration, we get By plugging into the previous equation to get its inner integral, we get The right-hand side of equation is found by reinterpreting the resultant contour integral in terms of a two-variable I-function. Integrals are derived in a similar fashion using the solutions to equations. Special Cases All exponents cancel at r=2, and 𝐴𝑗 (𝑗 = 1, . . ., 𝑝), 𝐵𝑗 (𝑗 = 1, . . ., 𝑞), 𝐶(𝑖) 𝑗 (𝑗 = 1, . . ., 𝑝𝑖, 𝑖 = 1 . . ., 𝑟), and 𝐷(𝑖) 𝑗 (1, . . ., 𝑞𝑖, 𝑖 = 1 . . ., 𝑟) Similar findings in H-function of two variables are obtained as the I-function of "r" variables reduces to H-function of two variables.

The theory of functions of several complex variables is the branch of mathematics dealing with functions The results presented in this paper generalize the results of H-function of ―𝑟‖ variables. The generality of the integrals is quite broad. It is possible to get many different specialized functions by appropriately specializing the I-function of two variables. Again, they may be reduced to other multivariate hypergeometric polynomials and classical appropriately specializing the coefficients. Thus, the equivalent conclusions involving a large number of simpler functions and polynomials would follow from our findings.

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Research Scholar Mathematics, Awadhesh Pratap Singh University, Rewa, Madhya Pradesh.