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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 $e$.

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    @MattBrenneman and everyone else! I've added the passage in the book. Hopefully it's everything you mentioned, and if not please feel free to add (or emphasize) any other parts!2012-02-21

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The natural exponential function is defined by $\exp(z) = \sum_{n=1}^\infty {z^n\over n!}.$ This is an entire function. It is not hard to show it has all of the expected properties.

To define $z^w$ you must define something like $z^w = \exp(z\log(w)).$

Unfortunately, the exponential function is $2\pi i$-periodic. Therefore it is not 1-1, so the business of defining a logarithm function becomes tricky. You must choose a domain to restrict the exponential function to so it is 1-1. And there the trouble begins. But where the trouble begins, complex analysis begins in all of its beauty and elegance.

I quote one of my grad school professors, Sidney Graham, who said, "There are those who say that the study of complex variables is the study of the logarithm function."

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The point is that we want "the" exponential function to be single-valued. If you want to write $\exp(1/n)$ as ${\rm e}^{1/n}$, that singles out one "$n$'th root of e". There are still $n$ $n$'th roots of e, it's just that only one of them is written as ${\rm e}^{1/n}$.