ANALYTIC EVALUATION OF CERTAIN CLASSES OF DEFINITE INTEGRALS INVOLVING SPECIAL FUNCTIONS

Ranjhan Ali; Israr Ahmed; Irshad Ali; Muhammad Arif; Soyam Kapoor; Manav Dev; Bisma Memon

doi:10.71146/kjmr859

Authors

Ranjhan Ali Department of Mathematics, Shah Abdul Latif University Khairpur, Sindh, Pakistan. Author
Israr Ahmed Department of Mathematics, Shah Abdul Latif University Khairpur, Sindh, Pakistan. Author
Irshad Ali MS Scholar, Department of Mathematics, Shah Abdul Latif University Khairpur, Sindh, Pakistan. Author
Muhammad Arif Crescent College of Accountancy Lahore, Punjab, Pakistan. Author
Soyam Kapoor Department of Computer System Engineering, IBA Sukkur University, Sindh, Pakistan. Author
Manav Dev Department of Computer System Engineering, IBA Sukkur University, Sindh, Pakistan. Author
Bisma Memon Department of Computer System Engineering, IBA Sukkur University, Sindh, Pakistan. Author

DOI:

https://doi.org/10.71146/kjmr859

Keywords:

Special Functions, Convergence Analysis, Analytic Methods, Mathematical Modeling

Abstract

It is an analytic discussion of definite integrals of special functions, and stability analysis, mostly with respect to convergence behavior. Some of the types of integrals that are analyzed in the paper are exponential, power-law, logarithmic, oscillatory and bounded integrals. Findings show exponential-type integrals converge at the fastest rate (up to about 92) because of the exponential damping effect of the decays. It also has good convergence rates of about 88 that is proven by the bounded integrals that it is stable at finite limits.

Conversely, convergence of power-law integrals (65%), conditions being found by the parameters and convergence of integration of logarithms at approximately 70% because of the sensitivity to the limits of the intervals. The rate at which the oscillatory integrals (58 percent) converge is the least and also it is accompanied with the difficulties of periodic oscillations. The parameter limits of improper integrals with infinity limits should be strict to guarantee convergence and should have a success rate of about 60%.

The study reveals that the parameter values, e.g., ppp, are significant in determining convergence. As an example, ∫ 0 infinity x p -1 -1 d x converges whereas ∫ 0-unit x -1 d x does not. These results highlight the significance of mathematical conditions in providing sound solutions.

The overall analysis suggests that the effectiveness of the methods used in analysis is really high with a high reliability of over 80 percent in the clear cases. It gives a very good idea of the choice of methods to be applied to find definite integrals, and the significance of convergence conditions in advanced mathematics.

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