Definition of exponential function, single-valued or multi-valued?

The notation $e^z$ is just shorthand for $\exp(z)$. Everybody agrees that when $e^z$ is evaluated it means $$ e^z=\exp(z)=\sum_{n\ge0}\frac{z^n}{n!} $$

Notation is not always consistent, unfortunately.

Indeed, if we would interpret $e^z$ as $\exp(z\log e)$, where $\log e$ is any determination of the logarithm, we'd get infinitely many values, unless $z$ is an integer. This is because $\log e$ can be any complex number $w$ such that $\exp(w)=e$ and it's easy to see that $w=1+2ki\pi$, for $k\in\mathbb{Z}$. Thus $\exp(zw)=\exp(z(1+2ki\pi))$. For instance, if $z=1/2$, we would have to assign $e^{1/2}$ both the values $\sqrt{e}$ and $-\sqrt{e}$.

However, as I said at the beginning, notation is a bit sloppy in this respect. Writing $e^z$ instead of $\exp(z)$ is deemed more practical and so $e^z$ used with the convention that it means the same as $\exp(z)$, mainly because single valued functions can be better manipulated algebraically.

As you note, people generally avoids using $a^z$ for $a\ne e$, when $z$ possibly varies in the complex numbers. In the case of positive real $a$, however, since $\log a$ has a well defined unique real value, there's no difficulty in defining and using $a^z=\exp(z\log a)$, where $\log a$ means that unique real value. The algebraic property $a^{z_1+z_2}=a^{z_1}a^{z_2}$ holds without restriction, with this convention (but only for positive real $a$).

• Indeed, unfortunately, the standard definitions of $e^z\;(z\in\mathbb C)$ as single-valued—which we generally want—are consistent, when using the logarithmic $(\log)$ definition of $e^z$, only with using the principal branch $\mathrm{Log}.$

  I.e., usually, $$e^z := 1+z+\frac{z^2}{2!}+\cdots \\= e^{z\,\mathrm{Log}(e)} \\≠ e^{z\log(e)};$$ however when wanting the function to be multi-valued (e.g. when determining $n$th roots), $$e^z := e^{z\log(e)}.$$

• The multi-valued definition of $e^z$ (i.e., using $\log$ instead $\mathrm{Log}$ to define $e^z$) results in $$e^{x+iy} = e^{x+iy}e^{2kπ(ix-y)}$$ being multivalued—which is generally not desired. For example, this apparently contradictory result ensues: $$e^{i\pi}=-e^{2n\pi^2}\\\neq-1.$$

• Until reading this post, I had thought that $\exp()$ and $e^{()}$ are synonymous; it turns out that the convention is that the former denotes the single-valued variant of the latter. It's indeed a good disambiguation.