This is the ninth book containing examples from the Theory of Complex Functions. We shall here treat the important Argument Principle, which . is applied in connection with Criteria of Stability in Cybernetics. Finally, we shall also consider the Many-valued functions and their pitfalls. Even if I have tried to be careful about this text, it is impossible to avoid errors, in particular in the first edition. It is my hope that the reader will show some understanding of my situation.