The world of complex Analysis is a fascinating and wonderful world of mathematics, more fascinating is the function of complex variables called Analytic function. Analytic function is our main focus in this work, we will be exploiting the effect of this function on geometric figures in the complex plane and what it does to physical systems.

Conformal mapping could be used to determine harmonic functions. The beauty here is that harmonic functions are solutions to the well-known Laplace equations in two dimensions. Most physical phenomena such as the steady state temperature distribution in solids, electrostatics, inviscid and irrational flow (potential flow) are mostly describe by Laplace equation. Therefore, conformal mappings can be effectively use for constructing solutions to the Laplace equation on complicated planar domains that appear in a wide range of physical problems, including fluid mechanics, aerodynamics, thermomechanics, electrostatics and electricity. Conformal mapping has proven to be a good engineering tool to solve footing on slope problems [9]. The complex velocity potential can be determine by solving a problem in either a horizontal or vertical stripe[8]. To clearly understand what conformal mapping does to geometric figures, we will briefly look at the celebrated Riemann Mapping theorem. Any simply connected domain in the complex plane, except the entire complex plane itself, can be mapped conformally onto the open unit disk. There are different aspects of conformal mapping that can be used for practical applications, the essence remain the same. Conformal mapping are optimal for solving various physical and engineering problems that are difficult to solve in their original form and in the given domain. Onyelowe [9] in their work  determined  how  the critical  stress  is spread  with  respect  to the  rupture  angle  using  a  conformal  transformation. In their words, a conformal transformation has proved to be a good engineering tool to solve footing on slope problems, and they concluded that the critical normal stress distribution of footing on a slope is spread evenly along the slip surface with the mapping technique. Lloyd [7] and Harri [4] presented algorithms for solving conformal mapping on a region or multiple connected domain and its implementation. These methods often employ graph theory and in particular, graph products well described in West [11] and Imrich and Klavzer [5] respectively. The complex velocity potential can be determined by solving a problem in either a horizontal or vertical strip [8].

Computer process stimulated appearances of many numerical conformal mappings construction methods. Many of these methods were connected with the integral equation solutions. If we want to map a given simple connected domain to the disk, then we solve a linear integral equation either analytically or Numerically [2,3,6,10]. In this study, we explore analytic functions when certain conditions are imposed on it. In particular, it aimed at elucidating the topic of conformal mappings. Various examples are given to show how conformal maps change given domains and help to solve some boundary-value problems, which are difficult to solve in their original domains.