  In this work, we spotted the light on one of the really important concepts
and turned it into 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.
