Simple confusion in complex analysis

Question

I've been learning Complex Analysis from George Cain's website: https://people.math.gatech.edu/~cain/winter99/complex.html

In chapter 3, Elementary Functions, it claims that the complex logarithm function is not single-valued, which I can understand.

However, I can't seem to understand how the complex exponential function exp(z) and complex power function $z^c$ are also multi-valued...

For example, at the end of the chapter, the reader is meant to feel unsatisfied when confronted with an expression like $e^{1/2}$. Its meant to be confusing since it may be suggesting two different numbers $\sqrt{e}$ and $-\sqrt{e}$.

I can't see how such exponentials and powers can be multi-valued.

Answer 1

The exponential function is not multivalued.

BUT:

We have two standard definitions:

$$\exp(z)=\sum_{n=0}^\infty\frac1{n!}z^n,$$

$$z^w=\exp(w\log(z)).$$

So $\exp(z)$ is definitely single-valued, while $z^w$ is multivalued.

The question is not whether the exponential function is multivalued; it's not, and the author did not say it was. The question is whether $e^z$ is multivalued!

At this point people object vehemently that $e^z=\exp(z)$, so no it's not multivalued. This raises the question "Is it true that $e^z=\exp(z))$?". And the answer to that question is "yes or no, depending".

In fact any time people see the notation $e^z$ they interpret that notation as meaning $\exp(z)$; this is of course a good thing because that's what the writer meant by the notation. But that's not consistent with the definition of $z^w$. The definition of $z^w$ says that $$e^z=\exp(z\log(e)),$$and that is multivalued!

If we want to say that $e^z=\exp(z)$ officially, by definition, then we need to modify the definition of $z^w$ to read $$z^w=\begin{cases} \exp(w\log(z)),&(z\ne e), \\\exp(w),&(z=e). \end{cases}$$

And that would be a really bad definition. For example, we couldn't say $z^w$ was multivalued any more, we'd have to say "$z^w$ is multivalued unless $z=e$". Similarly for anything else interesting we might say about $z^w$.

In most people's minds it's actually true that $z^w$ is multivalued except when $z=e$. But this is not for any mathematical reason; $\log(e)$ is multivalued just like any other logarithm. The reason people think $z^w$ is multivalued except for $z=e$ is just an artifact of the bad standard notation.

Summary

1. No, the exponential function is not multivalued, and nobody said it was; the quote from the author in question does not say $\exp$ is multivalued.

2. What people "always" mean by the notation $e^z$ is single-valued.

3. According to the actual definitions the notation $e^z$ does denote something multivalued.

Bad notation. Very bad. Too late to change it.

---

Not to snipe at the other users, but perhaps to clarify - I'm saying they're wrong, but we have no disagreement about the math, just about notation:

The other two answers I see argue that the exponential is single-valued, which is a straw man; the author didn't say it was multi-valued. They're missing the point, which you can see from locutions like "the exponential function $e^z$". They seem to be saying that the author is saying the exponential is multivalued, which again is missing the point, he never said that. The author's point is not mathematical, it's a valid point about bad notation; the definition of $z^w$ says that $e^z$ is not the exponential function, although standard convention says it is.

Answer 2

The exponential is not multi-valued. However, the power function is indeed multi-valued. The reason is that we may write

$$z^c = e^{c \log{z}} $$

so that, if $\log{z}$ is know only within a term of $i 2 \pi k$, $z^c$ is known only to within a factor of $e^{i 2 \pi c}$.

EDIT

The author of the treatise in questions states the following:

"Far more serious is the fact that we are faced with conflicting definitions of $z^c$ in case $z=e$. In the above discussion, we have assumed that $e^z$ stands for $\exp{z}$. Now we have a definition for $e^z$ that implies that $e^z$ can have many values."

Horsesh-t. The author is introducing a level of confusion over the definition of a variable. For example, $z^c$ is not multi-valued in $c$, so saying that any definition of the exponential is multi-valued is disingenuous at best. Please, forget this last paragraph - you will sleep better and be all the wiser.

Answer 3

The complex exponential function $e^z$ is not "multi-valued." The power function $z^c$ is multi-valued if $c \notin \mathbb{Z}$, because of how such a function is defined, $z^c = e^{log\ z^c} = e^{c\ log\ z}$ where the indeterminacy of the complex logarithm carries over to the power functions.

I think by convention $e^{\frac{1}{2}}$ should always be interpeted as $exp{(\frac{1}{2})}$, i.e. evaluate the exponential function before the fractional exponent. Then the fact that $(e^{\frac{1}{2}})^2 = (-e^{\frac{1}{2}})^2$ doesn't contradict $e^z$ being well-defined. The examples of $e^{\frac{1}{2}} = \sqrt{e}$ or $e^{\frac{1}{2}} = -\sqrt{e}$ arising from evaluating the fractional exponent first as if we were evaluating $z^{\frac{1}{2}}$ at $e$ would make a mess of the complex exponential.