Faber polynomials and theirs zeros

ABSTRACT This study entitled as " Faber polynomials and theirs zeros " consists of three chapters. In the first chapter which is a preliminary form to next ones, some important definitions and theorems concerning with the basic properties of Faber and generalized Faber polynomials were given. The Chapter 2 has seperated to be obtained Faber polynomials for some suitable subregions of complex plane as m - fold symmetric domains, circular lunes and annular sectors. Moreover, we derived a new determinant representation which relates the zeros of Faber polynomials to the eigenvalues of a certain matrix, and illustrated some numerical computations and various examples to find the zeros of Faber polynomials associated with m - fold symmetric domains. When it comes to chapter 3 that is main part of our study, it was expressed mapping function in terms of the focus of ellips and Faber polynomials were obtained for ellips with different focus. In addition, the relation between the zeros of generalized Faber polynomials and eigenvalues of Pn matrix has brought out and as result of this, it has been shown that the zeros of derivative of Faber polynomials can also be found using Pn matrix. Finally, the theorem as expressed with "If E is m -fold symmetric domain, then the set of zeros of generalized Faber polynomials are also m -fold symmetric" was proved with suitable choosing of the weighted function g(z). In particular, if we choose g(z) = 1 and g(z) = O'fc) in the Theorem 3.2.2, then it will be obtained Theorem 2.6.5 that has been proved by He M. [10] and it will be shown that the set of the zeros of derivatives of Faber Polynomials was m-fold symmetric too, respectively.