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.
[Math] Applications of complex exponential
applicationscv.complex-variablesmathematical-biologyphysics
Best Answer
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.