EVALUATION OF CERTAIN INTEGRAL TRANSFORMS AND THEIR APPLICATIONS TO SPECIAL FUNCTIONS

Abstract

It is a critical discussion of definite integrals with transformation of integration with special consideration being paid to the application of the integrals in special functions. Different kinds of integrals (exponential, trigonometric and mixed-function) were systematically analyzed. The results indicate that almost 80 percent of integrals solved and simplified to simplified algebraic equations with the help of Laplace and Fourier transforms and 20 percent were forced to employ alternative means of analysis. Analysis of convergence has found that almost three-quarters of integrals were discovered to be standard convergent, and a quarter of these required further conditions to be stable. The impact of the comparison of performance of the transform based algorithms with the traditional direct integration algorithms showed that the transform based algorithms cut the cost of computation by an order of 60. Other than this, the estimation of error has also been a criterion of correctness.oscillating or inappropriate integrals up to 90 percent. Also observed in the results are that transform methods are particularly particularly handy in solving integrals of special functions, e.g. Bessel functions, Gamma-type functions, etc., which are commonly encountered in problems of physical and applied mathematics. The paper confirms, as a rule, the existence of a highly efficient and stable system of the evaluation of the intricate definite integrals with the assistance of the integral transforms. The result also leads to the further development of methods of analysis and gives a practical implication to the procedures of the mathematical modeling, the engineering analysis, processes of scientific computations where the correct evaluation of the integrals is needed