Complex analysis is one of the classical branches in mathematics with roots in the 19th century and just prior. Important names are Euler, Gauss, Riemann, Cauchy, Weierstrass, and many more in the 20th century. Complex analysis, in particular the theory of conformal mappings, has many physical applications and is also used throughout analytic number theory. In modern times, it has become very popular through a new boost from complex dynamics and the pictures of fractals produced by iterating holomorphic functions. Another important application of complex analysis is in string theory which studies conformal invariants in quantum field theory. Holomorphic functions are complex functions defined on an open subset of the complex plane that are differentiable. Complex differentiability has much stronger consequences than usual (real) differentiability. Most elementary functions, including the exponential function, the trigonometric functions, and all polynomial functions, are holomorphic. For every student, teacher and researcher in the subject it offers a solid basis for an in-depth understanding of the entire subject area.