A given function $f$ from complex numbers to complex numbers, we can write it as

$$ f(z)= u(x,y) + iv(x,y) $$

where we call $u$ the real part, and the Imaginary part

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In Complex Numbers, we mostly work with the special case of functions like logarithms, and polynomials, and etc.

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for example, if $f(z) = z^2$, where $z \in \Complex$. then $f(re^{i\theta})=r^2e^{i2\theta}$ which can be written as

$$ f(r,\theta) = r^2cos(2\theta) + ir^2sin(2\theta) $$

where $u(r,\theta) = r^2cos(\theta)$ and $v(r,\theta) = r^2sin(\theta)$

Just as we cannot represent real-valued functions graphically in a single plane when dealing with higher dimensions, the same limitation applies to complex-valued functions. Instead of trying to plot them on one plane, we treat complex functions as mappings from one complex plane (the z-plane) to another complex plane (the w-plane).

For example, consider the function $w = f(z) = u(z)+iv(z)$

We can visualize how this function transforms a region in the z-plane into a corresponding region in the w-plane. This approach helps us understand the geometric behavior of complex functions, such as stretching, rotating, or folding regions of the complex plane.

A concept closely related to functions arises when we consider rules that assign more than one value to a given point $z$ in the domain. Such rules are often called multi-valued mappings or relations, rather than functions in the strict sense. These appear naturally in the theory of complex variables, just as they do in real analysis.

When dealing with such multi-valued relations, it is common to select one particular value in a consistent way over the domain (when possible), thereby defining a single-valued function known as a branch of the original relation. This process allows us to study these objects using the tools of complex function theory.

Example

Let $z$ denote any non-zero positive complex number. we know that $z^{1/2}$ has two values

$$ z^{1/2} = \pm\sqrt{r}\exp(i(\frac{\Theta}{2})) $$

where $r = |z|$ and $\Theta$ is principal value of $arg{z}$. but if we choose only positive value of $\pm\sqrt{r}$ and write

$$ f(z) =\sqrt{r}\exp(i(\frac{\Theta}{2})) $$

the function is well-defined on set of none-zero numbers