Bicomplex functions and some applications

In this thesis, bicomplex numbers and functions are discussed. The first section is the introduction section and this section includes the areas where bicomplex numbers are used and a literature review. In the second section, basic definitions and properties of bicomplex numbers are given and idempotent representations, modules, and conjugates of bicomplex numbers are emphasized. In addition, bicomplex linear spaces, elementary functions, zero divisors, and matrix representations are examined in this section. And bicomplex valued elementary functions and the concept of derivative of these functions are also examined in detail. In the third section, sequences and series of bicomplex functions are discussed and their integral formulas are given. Again in this section, the existence and properties of the most commonly used Laplace and Fourier transforms are discussed. In addition, the integral form defined to find the bicomplex and inverse bicomplex Laplace transform is examined in detail with the help of examples. Since it is quite difficult to find the inverse bicomplex Laplace transform in the integral form, another method, the convolution theorem, is also examined. In the same section, the existence and properties of bicomplex Fourier and Hartley transforms are discussed in detail. In the section, some examples related to bicomplex Fourier transforms are also given to reinforce the subject. As an application, the integral transform of the Bessel differential equation, which is encountered as one of the special differential equations, is discussed and its solution is given by using the properties of bicomplex Laplace, Fourier transforms and some basic functions. In the fourth section, the set of bicomplex numbers whose coefficients are selected from the Leonardo number series is defined. Using these defined numbers, the formation of Pascal and similar triangles is examined and some new identities are given with the help of these triangles.