Complex Functions

# Complex Functions

Our main object of study in complex analysis will be studying complex functions similarly to how real functions are studied in real analysis. We make a precise definition of a complex function below.

 Definition: Let $A \subseteq \mathbb{C}$. A Complex Function is a function $f : A \to \mathbb{C}$. The Domain of $f$ is $D(f) = A$ and is the set of all complex numbers for which $f$ is stated to be defined on. The Codomain of $f$ is $\mathbb{C}$. The Range of $f$ is $R(f) \subseteq \mathbb{C}$ is the subset of the codomain containing all complex numbers for which $f$ maps to.

Every real function $f : A \to \mathbb{R}$ (where $A \subseteq \mathbb{R}$) is a complex function.

For example, the following function $f : \mathbb{C} \to \mathbb{C}$ is a complex function:

(1)
\begin{align} \quad f(z) = \overline{z} \end{align}

The domain of $f$ is $\mathbb{C}$ since for all $z = a + bi \in \mathbb{C}$, $f(z) = f(a + bi) = \overline{a + bi} = a - bi$ is well-defined. The codomain of $f$ is also $\mathbb{C}$, and since for any $z = a + bi \in \mathbb{Z}$ we have that $w = a - bi \in \mathbb{C}$ is such that $f(w) = f(a - bi) = \overline{a - bi} = a + bi = z$, we see that the range of $f$ is $\mathbb{C}$.

Of course, there are more complicated complex functions that we could define. For example, consider the following function $f : \mathbb{C} \setminus \{ 1\} \to \mathbb{C}$ defined by:

(2)
\begin{align} \quad f(z) = \frac{z^2 - 2i}{z - 1} + |z|^3 \end{align}

This function is much more complicated than the previous one but it is indeed a complex function none the less.

In general, graphing complex functions $f : \mathbb{A} \to \mathbb{C}$ is rather difficult. This is because both the domain and range of $f$ are subsets of $\mathbb{C}$ and $\mathbb{C}$ can be thought of as $\mathbb{R}^2$ is some sense, so we can intuitively view $f$ as a function that maps $\mathbb{R}^2$ into $\mathbb{R}^2$ and an accurate graph of $f$ would require us to be able to see in four dimensions. Nevertheless, we will still examine complex functions by first extending some familiar real functions such as the trigonometric functions, the exponential function, and the logarithm function to be more general complex functions.