Counting theorems and inverse function theorems for analytic functions
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In this master's thesis, we discuss the counting and Rouche's theorems. These theorems are used to find the roots of complex analytic functions.Also, we investigate the existence of the inverse function of an analytic function. In other words, when we have a function from a domain set A of the function to its image set B, we discuss whether we can define an inverse function from B to A. We introduce the Lagrange inversion formula which is used to examine the inverse functions. Also, we apply this formula to some analytic functions to study the properties of the inverse functions, including Lambert's functions. This function has been used in quantum physics, fluid mechanics and biochemistry.