In calculus we learn about many applications of real exponentials like $e^x$ for bacteria growth, radioactive decay, compound interest, etc. These are very simple and direct applications. My question is are there any similar applications of the complex exponential $e^z$? In other words, are there any phenomena in the natural world (physics, biology, etc.) which are modeled by the complex exponential? I am aware that it surfaces in electromagnetism and signal processing, although it seems to be buried in the equations and therefore indirect.

$\begingroup$ I suspect something about logarithmic spirals would be one of vaious answers to this. $\endgroup$– Michael HardyJan 14 at 20:14

$\begingroup$ Dynamical systems with stable equilibria come to mind. $\endgroup$– Michael HardyJan 14 at 20:15
The earliest application is the Mercator projection which was introduced long before the complex exponential was defined in the way we define it nowadays. $z\mapsto e^z$ is considered as a map from the plane $C$ to the Riemann sphere $S$, where the plane is equipped with the usual metric, and the sphere with the spherical metric. Then $e^z$ is the inverse of the Mercator projection.
The map can be characterized by two properties: a) it is conformal, and b) meridians and parallels correspond to straight lines in the plane.
Discovered by Mercator in 1569, this was the second nontrivial example of a conformal map that was considered historically. The first one was the stereographic projection discovered in antiquity (but not known then to be conformal).
Referemce: Robert Osserman, Conformal mapping from Mercator to the Millennium.

$\begingroup$ Wow. So if you take the complex plane, apply $e^x$ to the real coordinate, wrap that around the cylinder (via a covering map where the imaginary part is the angle), take the cylindrical projection onto a sphere, and then project stereographically onto the plane, that's $e^z$! $\endgroup$– MaxJan 16 at 5:28

$\begingroup$ Another good reference: Osborne, Peter (2013): The Mercator Projections, doi:10.5281/zenodo.35392 $\endgroup$– user4503Jan 16 at 20:05

$\begingroup$ @user4503's reference, clickably: Osborne  The Mercator projections. $\endgroup$– LSpiceJan 16 at 23:43
Early applications of $e^{i\omega t}$ in the context of electromagnetism were understood as a mathematical device: the physical fields are real, and the complex exponential is a convenient method to implement trigonometric relations. The earliest application I know of where $e^{i\omega t}$ acquires a physical significance is by Erwin Schrödinger, who needed it to describe the time dependence of the electron wave function. He initially believed, or hoped, that this would eventually also turn out to be mathematical device. In a 1926 letter to Hendrik Lorentz he wrote:
"What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. $\psi$ is surely fundamentally a real function."
It did not work out that way, the $e^{i\omega t}$ factor is still with us, and indeed just this week we learned of a Experimental refutation of realvalued quantum mechanics under strict locality conditions. For more on this, see Schrödinger’s bewilderment – Quantum theory needs complex numbers.
_{ One might wonder, why not just work with standing waves of electrons, thereby eliminating the complex phase factor? There is a fundamental difference here between factors $e^{ikx}$ and $e^{i\omega t}$. The former can be reduced to a real sine or cosine by superposition with $e^{ikx}$. The latter can not, the reason being that superpositions of $e^{i\omega t}$ and $e^{i\omega t}$ are forbidden for electrons (such a superposition would couple a particle to antiparticle, violating charge conservation). There may exist chargeneutral particles which are their own antiparticle (socalled Majorana fermions), and indeed, in that case a real wave equation applies. Suggestion for further reading: A road to reality with topological superconductors. }

3$\begingroup$ This is not really a "complex exponential", since $\omega t$ is real. Complex exponential is an entire function defined for all complex values of $z$. But I vote this up, since this application is by far the most important one:) $\endgroup$ Jan 14 at 22:14

1$\begingroup$ Yes, I found the question confusing too at first, but I think the intended question was something like, what applications are there of $e^{x+iy}$ other than the special cases when $x=0$ or $y=0$? Otherwise almost any appearance of $\sin x$ or $\cos x$ could be argued to be a complex exponential. $\endgroup$ Jan 15 at 6:01

$\begingroup$ @Timothy in my answer the Hamiltonian can have both real elements and imaginary elements so I cover the case in which neither x nor y is 0 (as well as many other cases). $\endgroup$ Jan 15 at 7:01

1$\begingroup$ @TimothyChow  I added a brief paragraph on why $e^{i\omega t}$ in the electron wave function cannot be simply reduced to a real sine or cosine. $\endgroup$ Jan 15 at 10:28
In quantum mechanics, the wavefunction $\psi(t)\rangle$ of a system with a constant Hamiltonian $H$, evolves according to:
$$\tag{1}\label{1}\lvert\psi(t)\rangle = e^z \lvert\psi(0)\rangle,$$ where $z$ is the complex matrix:
$$\tag{2}\label{2} z = \frac{\textrm{i}}{\hbar}Ht. $$
This is simply because in quantum mechanics, the wavefunction evolves according to the Schroedinger equation:
$$ \frac{\textrm{d}}{\textrm{d}t}\lvert\psi(t)\rangle = \frac{\textrm{i}}{\hbar}H \lvert\psi(t)\rangle\tag{3}\label{3}, $$
and Eq. \ref{1}, with $z$ chosen according to Eq. \ref{2}, is the solution to Eq. \ref{3}.
Currently, the very popular feature engineering technique in deep audio processing [audio classification, speaker recognition, noise suppression, speech enhancement] is to feed neural networks with spectrograms instead of raw waveforms. Given an audio waveform $x[n]$, the spectrogram is obtained by applying Discrete Short Time Fourier Transform (STFT) to $x$ by the formula $$\mathbf{STFT}\{x[n]\}(m,\omega)\equiv X(m,\omega) = \sum_{n=\infty}^{\infty} x[n]w[nm]e^{j \omega n},$$ where $w$ (not $\omega$) is some window function. In the essence, it is a new complex valued signal in timefrequency domain. This channels (real and complex, or absolute/power) are then fed to neural networks (usually 2D CNNs)

$\begingroup$ Lest anyone else wonders  from the linked Wikipedia page I gather that the function $w$ in the (sort of) convolution is the "smoothing window", and $j=\sqrt{1}$ (I never saw that notation before beyond the context of quaternions). $\endgroup$ Jan 16 at 23:05

$\begingroup$ @JulesLamers Yeah, It can be just rectangular window, but usually one adds additional smoothing. Here you can browse various windows: en.wikipedia.org/wiki/… The notation is consistent with the formula in my answer. Regarding $j$, I saw this notation few times; one interesting fact is that in python imaginary unit is in fact denoted by
j
. Thus, if you write in python1.0 + 1.0j
it will treat it as $1+i$. $\endgroup$ Jan 16 at 23:19 
$\begingroup$ Sorry, I just meant that $w$ is the 'window', $\omega$ a variable (as the notation suggests) $\endgroup$ Jan 16 at 23:21

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3$\begingroup$ I've heard engineering students refer to them as the "jimaginary numbers." $\endgroup$ Jan 17 at 3:18