Non-Sinusoidal Harmonics and Special Functions
This chapter shows that the harmonics were sinusoidal basis functions for a large, infinite dimensional function space. It aims to extend these ideas to non-sinusoidal harmonics and explore the underlying structure behind these ideas. The chapter reviews the sine and cosine functions in the Fourier trigonometric series representations as basis vectors in an infinite dimensional function space. It considers various infinite dimensional function spaces. Functions in these spaces would differ by their properties. A function that often occurs in the study of special functions is the Gamma function. Bessel functions arise in many problems in physics possessing cylindrical symmetry, such as the vibrations of circular drumheads and the radial modes in optical fibers. They also provide us with another orthogonal set of basis functions. The first occurrence of Bessel functions was in the work of Daniel Bernoulli on heavy chains.