TITLE: Applications of complex exponential
QUESTION [19 upvotes]: 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.

REPLY [4 votes]: 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)