The usual way of generating functions apply to complex variables as well as to real variables. By combining sums and products one arrives at polynomials, which are the simplest functions to define. By taking limits, polynomials can be extended to infinite series. Trigonometric functions, logarithms and exponentials arise from the process without much difficulty. In another direction, the inclusion of division among the constructing operations leads to rational functions and eventually to quotients of series and series of quotients.

- one-to-one and invertible
- Möbius transformations represented as cross ratios
- Möbius transformations representable as 2x2 matrices
- eigenvalues and eigenvectors of a Möbius transformation
- hyperbolic, parabolic, elliptic transformations