Step 1 :

The exponential function of a complex variable

Here we define and study extensively the extention of the real exponential function to the complex exponential function.  We look at its many important properties and in the process go through how the celebrated  number was constructed.

Step 2 :

The trigonometric functions of a complex variable

We use the complex exponential function to define the basic complex trigonometric functions and examine some of the rules from real numbers that remain intact.  On the other hand, we highlight some of the rules that change with the complex case.

Step 3 :

The logarithmic function of a complex variable

Finally, in this study of complex elementary functions, we define the complex  logarithmic function (“inverse” of the complex exponential function) and we note a major change in the complex form in that the complex  logarithmic function is a multi-valued function.