Computational & Functional Analysis of Special Functions with Arbitrary Parameters

This work presents a comprehensive computational and functional analysis of special functions, specifically focusing on cases involving arbitrary integer parameters. Using integral transformations and identities, such as those from the Beta, Gamma, poly-gamma, and Zeta functions, we explore and derive solutions to various complex integral expressions. The problem sets address combinations of logarithmic, trigonometric, and exponential functions, including of the form ln(x) tan( x b ) and arcsinh(csch(mx)), where b, m ∈ Ζ. Each solution is derived under generalized conditions, allowing for a range of integer parameter values. The study demonstrates the use of advanced mathematical techniques, including substitution, binomial expansions, and Fourier series, to simplify and compute the integrals. The results offer insights into the computational strategies required for complex special functions and serve as a reference for future explorations of such functions in both theoretical and applied mathematics.