Trigonometric Krätzel Functions: Properties, Integral Transforms, and Applications in Diffraction Phenomena

ABSTRACT This paper introduces two new mathematical functions, the sine Krätzel function and the cosine Krätzel function, which extend the classical Krätzel function into the trigonometric domain. These functions are rigorously defined through integral representations and their fundamental properties, such as absolute convergence, generating functions, and derivative formulas, are investigated in detail. The integral transforms of these functions, including Mellin, Fourier, and Laplace transforms, are derived, highlighting their analytical flexibility. Furthermore, the study explores the applications of these functions in wave optics, specifically in the context of Fresnel and Fraunhofer diffraction patterns, demonstrating their utility in understanding light diffraction phenomena. Numerical examples and graphical visualizations are provided to illustrate the influence of key parameters on diffraction patterns, emphasizing the potential of these functions in applied mathematical and physical research.