Complex Exponentials

The complex number $\tilde{z}=a+bi$ has a real part $a$  and an imaginary part $b$, corresponding to its projection on the real or imaginary number lines. The first step in understanding complex exponentials is to think of a complex number as a vector. You could think of it as a rectangular vector $\textbf{z}=(a,b)$. You could also think of it as a polar vector $\textbf{z}=(r,\theta)$, where $r=\sqrt{z_{\mathrm{re}}^2 + z_{\mathrm{im}}^2}$ and $\theta=\arctan\frac{z_{\mathrm{im}}}{z_{\mathrm{re}}}$. From here, you can apply Euler’s formula:

$$\label{eq:euler} e^{i\theta}=\cos\theta + i\sin\theta$$

Some trigonometry will tell you that this exponential represents a complex unit vector (i.e. a vector with magnitude 1) in the $\theta$ direction. You can therefore represent any complex vector in polar space as:$$\textbf{z} = r e^{i\theta}$$This notation is called a complex exponential, and it’s very useful for what we’re about to do. You see, complex exponentials behave almost exactly like sine waves. The magnitude $r$ of the complex exponential corresponds to amplitude, and the angle $\theta$  of the complex exponential corresponds to phase.

I’ll give you one example of the usefulness of complex exponentials. Let’s say you want to add or multiply some sine waves with different amplitudes and phases. Combining trig functions can get really messy.The wavenumber and the phase are stuck together inside the cosine function, and the amplitude is completely separate. So let’s use Euler’s formula to put it into complex exponential form:$$A\cos(kx+\phi) = Ae^{i(kx+\phi)}$$Yes, I’m playing fast-and-loose with my equals signs—technically, the cosine is only the real part of the complex exponential. But it turns out that you can actually hold off on taking the real part until the very very end, so it’s easier in practice to just say that the wave is the same as the complex exponential. In this form, we can easily separate the phase from the rest of the wave argument:$$Ae^{i(kx+\phi)} = Ae^{i\phi}e^{ikx}$$The $e^{ikx}$ term will change with $x$, but $Ae^{i\phi}$  is a constant, so let’s lump them together into a complex vector defined as:$$\tilde{A} \equiv Ae^{i\phi} = (A,\phi)$$We now have a complex amplitude that contains both the amplitude and the phase information. These complex amplitudes are much easier to work with, particularly for computers.

For a much more in-depth explanation of complex exponentials, the University of Wisconsin-Madison has published this document, which includes some homework-style practice problems as well.