General Master Theorems of Integrals with Applications

Abstract

Many formulas of improper integrals are shown every day and need to be solved in different areas of science and engineering. Some of them can be solved, and others require approximate solutions or computer software. The main purpose of this research is to present new fundamental theorems of improper integrals that generate new formulas and tables of integrals. We present six main theorems with associated remarks that can be viewed as generalizations of Cauchy’s results and I.S. Gradshteyn integral tables. Applications to difficult problems are presented that cannot be solved with the usual techniques of residue or contour theorems. The solutions of these applications can be obtained directly, depending on the proposed theorems with an appropriate choice of functions and parameters.

A series representation for Riemann's zeta function and some interesting identities that follow

“…Here, we are interested in such a Theorem recently introduced by Glasser [1] and studied by Glasser and Milgram [5], [6], where many specific and general examples were given. Considering the importance of the subject, it is worth noting that a survey of other "Master" theorems was included in [5]; further citations to the literature on this subject can be found in [7], [8], [9] and [10] where other, newly derived "Master" theorems are discussed, and their variations explored.…”

Mastering Complex Integrals in Multidimensional Spaces:Generalized Theorems and Broad Applications

“…This section aims to clarify the fundamental concepts, definitions, and lemmas that underlie our comprehensive research. Building upon and expanding the work in one dimension from [5,6,8], this research delves into multidimensional parameters in the complex plane. This approach is crucial for comprehending the more advanced and nuanced aspects of complex analysis.…”

A Fundamental Criteria to Establish General Formulas of Integrals

“…In this section, we present new master theorems to help mathematicians, engineers, and physicists solve complicated improper integrals. To obtain our goal, we present some facts about analytic functions [8,10,[13][14][15]. Assuming that f is an analytic function in a disc D centered at α, then using Taylor's expansion, where α, β, and θ are real constants, we have…”

“…Many applications need improper integrals to handle, either in the calculations or in expressing the models, especially in engineering, applied mathematical physics, electronics engineering, etc. [9][10][11][12][13][14][15]. Some of these integrations can be solved simply, but others need difficult and long computations.…”