Relationship between mathematical and physical constants

Euler's formula ($e^{\pi i}=-1$) is used to obtain a relationship between mathematical ($\pi$, $e$, $i=\sqrt{-1}$) and physical ($\varepsilon$, $h$, $c$) constants:
$$ \alpha^{-1}=\frac{hc}{\varepsilon^2}= \frac{e^{2\pi}+2e^{-\pi}+4\pi}{4}=137,0361134938\dots. $$