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This thesis looks at various problems relating to the value distribution of certain discrete potentials. Chapter 1 - Background material is introduced, the motivation behind this work is explained, and existing results in the area are presented. Chapter 2 - By using a method based on a result of Cartan, the existence of zeros is shown for potentials in both the complex plane and real space. Chapter 3 - Using an argument of Hayman, we expand on an established result concerning these potentials in the complex plane. We also look at the consequences of a spacing of the poles. Chapter 4 - We extend the potentials in the complex plane to a generalised form, and establish some value distribution results. Chapter 5 - We examine the derivative of the basic potentials, and explore the assumption that it takes the value zero only finitely often. Chapter 6 - We look at a new potential in real space which has advantages over the previously examined ones. These advantages are explained. Appendix - The results of computer simulations relating to these problems are presented here, along with the programs used.
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