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In this book, we spotted the light on one of the really important concepts and turned it to a mathematical branch instead of separate equations studied individually in different specializations of science. The existence and uniqueness of solutions for complex differential equations have been proved with many mathematical generalized tools. Homotopy perturbation method has been used and implemented as a method for solving linear complex differential equations with which is the first time such a method used to solve an equation in the complex plane. An analytical method of solutions has been investigated deeply with complex series solutions have generalized also using Laurent series expansions. Some really important application in engineering and complex geometry has been implemented with deep understand- ing, such as Airfoil application for airplane and spaceships wings and the fans of jet engines, and the Schwarz-Christoffel transformation.

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