ABSTRACT

Most real life problems in engineering, physics and hydromcchanics are represented mathematically by function of a complex variable, hence difficult to solve. The Cauchy-Riemann Equations are satisfied in a region R where it is supposed that the partial derivatives are continuous in R for a function of a complex variable analytic in R. Moreover, this Cauchy-Riemann Equations has a vital application to the solution of two dimensional problems by the separation of any analytic function of z into its real and imaginary parts. In this thesis, I. have run from the theory of complex variable an integral part of mathematics appropriate to the physical application involving developing a large amount of groundwork in analysis such as definition of continuity and differentiability through the derivation of Cauchy-Riemann Equation using the necessary and sufficient conditions for analyticity of a function and its representations in pillar coordinates. Consequently, the few number of worked examples given in the text, are all of the mathematical type. One of the many applications of the Cauchy-Riemann Equations is the Harmonic and Conjugate functions. Hence, Cauchy-Riemann Equation is necessary and sufficient. Okoko