Two planes
In this section we present some of the commonly known methods for visualizing complex functions. We will also show a couple of examples of their use. Our goal for now is just to briefly introduce these methods. They all have their advantages, drawbacks, application areas. But perhaps none of them can give such complete, concise and perspicuous images like the 'colorful' method.
The first method may be called the method of two planes because it basicly uses two planes: the so-called z and w planes, and examines where certain points and shapes (lines, line segments, circles) of the z plane are mapped and plots these on the w plane. (If we would agree that we always examine the same set of points every time, then one plane would also be sufficient.)
Let us consider first the square function. It can be shown that the lines parallel to the imaginary axis are mapped to parabolic curves.
\(f(z) = z^2\)
We have plotted four lines on the z plane, of which the image of two coincides resulting only in three parabolae. The following short calculation could help to see this. Fix a real number \(a_0\) and let \(b\) take real values. So: \[ (a_0 + b i)^2 = \underbrace{-b^2 + a_0^2}_{u} + \underbrace{2 a_0 b}_{v} \cdot i,\qquad u = -\frac{1}{a_0^2} v^2 + a_0^2. \]
Our second example is the exponential function. Let us examine where the lines parallel to the real and imaginary axis are mapped.
\(f(z) = \exp\,z\)
The images of these sets are respectively half-lines starting at the origin and concentric circles with their center also at the origin. These facts are explained by the calculations below based e.g. on Euler's formula: \[ \exp\,z = e^z = e^{x+iy} = e^x \cdot e^{iy} = e^x (\cos\,y + i\,\sin\,y). \]
When we fix one of x and y and move the other one along the real numbers then the formulas read as parameterizations of half-lines and circles. (Not just the images of the horizontal lines shown on the z plane are plotted, but more.)