Visualizing the Exponential Function

Section 2.9 Visualizing the Exponential Function

Using Euler’s formula Section 2.6, we now know how to understand the exponential function for both real variables \(e^x\) and for pure imaginary variables \(e^{iy}=\cos y + i\sin y\text{.}\) What happens when we put these ideas together? We’d like to extend the definition of the exponential function to be valid when the independent variable is an arbitrary complex number \(z=x+iy\text{.}\) In particular, let’s extend the definition so that common exponential rule

\begin{equation} e^{z_1 + z_2}=e^{z_1}e^{z_2}\tag{2.9.1} \end{equation}

still holds. (Extending the domain of a function from the reals to the complexes is called analytic continuation, see Section 2.10).

Definition 2.11. The Exponential of a Complex Number.

Using these properties, we see that

\begin{align*} e^{z} \amp = e^{x+iy}\\ \amp = e^{x}\, e^{iy}\\ \amp = e^{x}\left(\cos y + i\sin y\right)\\ \amp = e^{x}\cos y + ie^{x}\sin y\text{,} \end{align*}

The exponential of a complex number is also a complex number; the real and imaginary parts are products of real exponentials and real cosines and sines.

To graph a complex-valued function of a complex variable, we would need to live in four dimensions: two for the input variable (domain) and two for the output variable (range). A common way to handle this problem is to plot the real and imaginary parts of the output on separate graphs, so that, in each case we only need two dimensions for the input variable and one for the output. The following activity explores such graphs for the real part of the exponential function \(\mathrm{Re}\, e^z=e^x\cos y\text{.}\)

Activity 2.5. Visualizing the Exponential Function of a Complex Variable.

Look at the figure below. The slider on the left-hand plot chooses various straight-line cross-sections of the complex-plane which range from the positive \(x\)-axis when \(\theta=0\) to the negative \(y\)-axis when \(\theta=-\pi/2\text{.}\) Start by moving the slider and seeing what happens to the green line segment in the right-hand plot while it is still in its initial position; this shows you the domain.

Next, click and drag on the right-hand plot (up and to the right) to rotate the graph until it looks three-dimensional. Now, on the right-hand graph, you are seeing the full graph of the real part of the exponential function \(\mathrm{Re}\, e^z=e^x\cos y\text{.}\) Move the slider back and forth between its two limiting cases: the real-valued exponential function \(e^x\) when \(y=0\) to the real-valued trigonmetric function \(\cos y\) when \(x=0\text{.}\) The left-hand graph shows just a graph of the cross-section, so its easier to see.

Between the two limiting cases, notice how the cosine function \(\cos y\) creates wiggles and the real exponential function \(e^x\) makes the magnitude of the wiggles increase for larger values of \(x\text{.}\)

Figure 2.12. The graph of \(\mathrm{Re}\, e^z=\mathrm{Re}\, e^{x+iy} =e^x\cos y\text{.}\)

Prev Top Next