Applications of complex exponential 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.
 A: 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}.
A: 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
$$\mathbf{STFT}\{x[n]\}(m,\omega)\equiv X(m,\omega) = \sum_{n=-\infty}^{\infty} x[n]w[n-m]e^{-j \omega n},$$
where $w$ (not $\omega$) is some window function. In the essence, it is a new complex valued signal in time-frequency domain. This channels (real and complex, or absolute/power) are then fed to neural networks (usually 2D CNNs)

A: 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 Gerard Mercator* in 1569, this was the second non-trivial 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).
*Not to be confused with his children Arnold and Rumold, also cartographers, or with
the mathematician Nicholas Mercator, a contemporary of Newton.
Reference: Robert Osserman, Conformal mapping from Mercator to the Millennium.
A: 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 real-valued 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 charge-neutral 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.

