Complex numbers are not just useful when solving polynomial equations, but they play an important role in many fields of mathematical analysis. With the help of complex functions, transformations of the plane can be expressed, solution formulae for differential equations can be obtained, and matrices can be classified. Not least, fractals can be defined by complex iteration processes. In this section we introduce complex numbers and then discuss some elementary complex functions, like the complex exponential function. Applications can be found in Chaps. 9 (fractals), 20 (systems of differential equations) and in Appendix B (normal form of matrices).