Power Series Functions: Logarithm and Exponential

This lecture delves into power series functions, focusing on the logarithm and exponential functions. The instructor explains the properties and convergence of the logarithm as a power series, its evaluation on specific elements, and its extension to the entire plane. The lecture also covers the exponential function, its radius of convergence, and its divergence outside a certain disk. Additionally, the concept of entire functions in Cp is introduced, along with a lemma on finding roots of such functions. The discussion highlights the uniqueness and properties of locally analytic functions, emphasizing the relationship between entire functions and polynomials.