4.3. Complex exponentials

Recall the definition of the exponential function as an infinite summation:

If you need a refresher on exponentials, see the appendix.

\[e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}\]

We normally think of this definition in terms of real-valued \(x\), but the idea carries over to complex exponentials \(e^z\) for \(z \in \mathbb{C}\) using exactly the same formula with \(z\) in place of \(x\).

Because any complex \(z\) is the sum of its real and imaginary parts, it can be helpful to separate the exponential using the product rule:

\[e^z = e^{a + \mathrm{j}b} = e^a \cdot e^{\mathrm{j}b}\]

Since \(a\) is real, we already have a good handle on how \(e^a\) behaves. Let’s focus on what happens just to that second factor, where the quantity in the exponent is purely imaginary: \(e^{\mathrm{j}b}\).

Fig. 4.6 shows what happens as we form better approximations to \(e^{\mathrm{j}b}\) by taking more terms in the summation (up to 50), for \(b \in [-4\pi, +4\pi]\). For clarity, the numbers in the interval \([-\pi, \pi]\) are highlighted.

Once Loop Reflect