THE MELLIN INTEGRAL TRANSFORM AND ITS APPLICATIONS

This article is devoted to the study of the Mellin integral transform and its applications in modern mathematical analysis. The paper examines the theoretical foundations of the Mellin transform, its fundamental properties, and its relationship with other classical integral transforms. Particular attention is paid to the role of the Mellin transform in solving problems involving scale invariance, asymptotic behavior, and multiplicative convolution. The study demonstrates that the Mellin transform provides an efficient analytical framework for evaluating improper integrals, analyzing special functions, and simplifying complex mathematical expressions. In addition, the article highlights practical applications of the Mellin transform in mathematical physics, signal and image processing, algorithm analysis, and other applied fields. The results confirm that the Mellin integral transform is a versatile and powerful tool that bridges theoretical and applied mathematics, offering broad potential for further research and development.