In Churchill's book on complex variables, the $n^{th}$ root of $e$ is defined to be $e^{1/n}$. A comment is made that in this respect $e$ is treated differently than the $n^{th}$ roots of other complex numbers (in the sense that there are typically n roots of the nth root of a number in complex analysis rather than just one as in the case of $e$).
I am curious why $e$ is treated so differently. Is there an obvious reason/motivation why?
Edit: The section from Churchill is,
As anticipated earlier, we define here the exponential function $e^z$ by writing $ e^z = e^xe^{iy}\ \ \ \ \ \ (z = x + iy)\ \ \ \ \ \ \ \ \ (1)$ where Euler's formula $ e^{iy} = \cos y + i\sin y$ is used and $y$ is to be taken in radians. We see from this definition that $e^z$ reduces to the usual exponential function in calculus when $y=0$; and, following the convention used in calculus, we often write $\exp z$ for $e^z$.
Note that since the positive $n$th root $\sqrt[n]{e}$ of $e$ is assigned to $e^x$ when $x = 1/n$ ($n = 2,3,\ldots$), expression (1) tells us that the complex exponential function $e^z$ is also $\sqrt[n]{e}$ when $z = 1/n$ ($n = 2,3,\ldots$). This is an exception to the convention that would ordinarily require us to interpret $e^{1/n}$ as the set of $n$th roots of $e$.