The Mean Value Theorem for Functions on the Complex Plane

by Vincent Peichel
Faculty advisor: Dr. Nasser Dastrange

In this project, we explore the Mean Value Theorem and Rolle’s Theorem as they apply to the complex plane. In the first section, we present examples that show that Rolle’s Theorem fails in the complex plane. Following this, we introduce a special form of Rolle’s Theorem that does work for analytic functions, namely the Complex Rolle’s Theorem. An example illustrates the concept of the Complex Rolle’s Theorem. Then, we present the Complex Mean Value Theorem and show how it works for an analytic function. Finally, we show how Flett’s Theorem fails in the complex plane.