Bridging Classical Theory and Modern Applications with Ruscheweyh Type Harmonic Functions

Abstract

Harmonic functions, which satisfy Laplace's equation, are integral to mathematics, particularly in potential theory, complex analysis, and the study of partial differential equations (PDEs). This paper delves into generalized Ruscheweyh type harmonic functions, which extend classical harmonic functions through a more adaptable differentiation approach. The Ruscheweyh derivatives, characterized by the generalized parameter α and order n, offer a framework for analyzing complex behaviors beyond traditional derivatives. The paper investigates the theoretical foundations of Ruscheweyh type harmonic functions, with a focus on their applications in modeling potential fields and solving generalized boundary value problems. Python code is employed to visualize harmonic functions and compute Ruscheweyh derivatives, showcasing the practical application of these theoretical ideas. This study underscores the versatility of Ruscheweyh type functions in complex analysis and mathematical physics, highlighting their utility in addressing more intricate scenarios and providing deeper insights into potential theory and function behavior. Future research will explore higher-dimensional applications, enhance computational methods, and integrate these functions with contemporary techniques in applied and theoretical research.