In the following chapters, we will see that, in order to define complex functions and their features, we can extend some properties of real functions; but, in some cases the mere extension will not be enough.

We start with the following:

Consider . A complex-valued function is a function:

We see that the function can be decomposed as:

where the functions are real-valued functions. In particular, we define:
Alternatively, we can use the isomorphism between and and look at and as two variables real functions:

In fact, by calling , we have that:

Now we define some complex functions as extensions of known real functions:

Trigonometric Functions: We recall that, thanks to the Euler's formula, we can write:

So, the natural extension to the complex field is:
Analogously, we have that:
by setting we see that:

We underline that All the complex exponentials of the form are called phases and have unitary modulus.

The following properties (simple to prove) are valid:

Logarithm function in : In order to define the complex logarithm, we use the polar form of complex numbers. So, we have:

Actually, we have to keep in mind that complex numbers are defined with a certain ambiguity in their polar form, so we have that:

The following properties are true: