New Approaches and Results in the Theory of Discrete Analytic Functions

1976 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. 1 Acknowledgements It. is my pleasure.to express my warm thanks and deep appreciation to my advisor, Harry Dym. for his guidance and constant friendly encouragement. I would also like to thank Miriam Drory who didn't save any effort to decipher m y .hieroglyphic handwriting and turned it into a beautiful typescript. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6. Binary operations in the set of solutions of a partial difference equation, Submitted. [Appendix A2]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A TECHNICAL REMARK "Formula (m,n)" means "Formula (m,n) in the present chapter". If referfence will be made to formulas from another chapter, then it will be preceded by the relevant chapter's number; e.g., 1 (3.2) means: formula (3.2) in chapter I. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INTRODUCTION This thesis is concerned with discrete analytic functions 2 A function f: Z £ from the two dimensional lattice to the complex numbers is said to be discrete analytic if for o every (m,n) € Z If we embed Z in C by identifying it with the set of Gaussian integers {m + in} we observe that (1) is an 'analyticity' condition: the difference quotients along the two diagonals of each unit square arc the same. Equivalently, discrete analytic functions can be characterized as the solutions of the homogeneous partial difference equation The theory of discrete analytic functions was initiated by Jacquelline Ferrand-Lelcng [13] and further developed by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the proofs of most of the classical theorems of Complex Aanalysis do not carry ^over directly to the discrete case because they \ rely heavily upon the fact that the class of analytic functions is closed under pointwise multiplication. This is not the case for the class of discrete …