Archimedes & Euler

A complex function that is its own derivative normalized to one at zero implicitly defines the famous Archimedean and Euler constants of circular motion and exponential growth. Even in a world of strong gravity, where the ratio of a circle’s circumference to its diameter noticeably varied from place to place, this exponential function and the axioms of mathematics would generate these same transcendental numbers.

Assuming the Taylor series expansion

f(z) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} z^n,

where z = x + i y \in \mathbb{C} with x,y \in \mathbb{R} and i = \sqrt{-1}, the derivative condition

f^\prime(z) = \sum_{n=1}^\infty \frac{f^{(n)}(0)}{(n-1)!} z^{n-1} = \sum_{n=0}^\infty \frac{f^{(n+1)}(0)}{n!} z^n \equiv f(z)

coupled with the normalization condition

f^{(n+1)}(0) = f^{(n)}(0) = f^{(n-1)}(0) = \cdots = f^{(1)}(0) =f^{(0)}(0) = f(0) \equiv 1

implies

f(z) = \sum_{n=0}^\infty \frac{z^n}{n!} \equiv \exp z.
Complex exponential repeatedly maps horizontal strips (left) to the entire complex plane (right), thereby defining the Archimedes and Euler constants

Complex exponential repeatedly maps horizontal strips (left) to the entire complex plane (right), thereby defining the Archimedes and Euler constants. Click for a better view.

Numerically plot this expression to discover two jewels. As in the figure, the exponential function maps the imaginary axis to the unit circle f(i\mathbb{R}) \rightarrow \mathbb{S}, with negative real parts forcing complex numbers inside (red) and positive real parts forcing complex numbers outside (blue). The function is exponential on the real axis with e-folding time 1, so f(1)=e, but periodic on the imaginary axis with period \tau, so f(z+i\tau)=f(z). Specifically, as it maps 1 to e, it maps horizontal strips of height \tau = 2\pi onto the entire complex plane, where

\pi =3.14159265358979323846264338327950\ldots

and

e = 2.71828182845904523536028747135266\ldots

are associated with Archimedes and Euler.

About John F. Lindner

John F. Lindner was born in Sleepy Hollow New York and educated at the University of Vermont and Caltech. He is a professor of physics and astronomy at The College of Wooster. He has enjoyed multiple yearlong sabbaticals at Georgia Tech, University of Portland, University of Hawai'i, and North Carolina State University. His research interests include nonlinear dynamics, celestial mechanics, and variable stars.
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