Branch cut of $\log(z)$ in comparison with branch cut of a function like $\sqrt{z}$

Question

I am looking at the explanation of the branch cut for $\log(z)$ defined at $ -\pi$ $< \theta \leq \pi$ and the explanation seems to be that the limit does not exist on the negative real axis because there is a jump of $2\pi$ as we cross the negative real axis and because we can approach the negative real axis from two different angles $-\pi$ and $\pi$. So in this case, say $z = -1$ then our limit does not exist because of the two different angles $-\pi$ and $\pi$. Then is it the argument of $z$ that makes this problematic, but where is the multivaluedness of the function, is it the two different angles $+2k\pi$ for each one where $k$ is an integer?

Then I observe a function like $ f(z) = \sqrt{z}$ and it is multivalued for example at $\pi$ because $f(z)$ = $\sqrt{2i}$ and at $3\pi$ because $f(z)$ = $-\sqrt{2i}$. We could also use the same angles as with $\log(z)$ and we would still get $\sqrt{2i}$ and $-\sqrt{2i}$. So my question is, for $\log(z)$, my text described it as the limit does not exist, but is it also multivalued? For $ f(z) = \sqrt{z}$, it's multivalued, but does the limit also not exist unless we cut out the negative axis, (assuming it's initially defined at $ -\pi$ $< \theta \leq \pi$. I am trying to determine the reasons for the branch cut for both functions (is it the limit not existing, the multivaluedness, or both)

Yes, there is one way to look at $\log z$ as multi-valued. Even the name “branch cut” is a hint at this - we are pruning away multi-valued cases. We often choose to do so with the real non-negative numbers is a choice of how we prune. We could choose to make the cut at $i\mathbb R^{\geq 0}$ or even something perverse like the set of points $${re^{ir}\mid r\in\mathbb R{\geq0}}$$ — Thomas Andrews, May 06 '21 at 20:42
$\log(-1)$ $=\pi$ and $\log(1)$ = $=\pi$ is an example of multivaluedness right — Elli, May 06 '21 at 20:48
In the hope that pictures of multi-valuedness of log and multi-valuedness of the square root are worth a thousand words.... — Andrew D. Hwang, May 06 '21 at 21:17
No, $\log(-1)=\pi(2k+1)i$ and $\log(1)=2\pi ki$ for $k$ integers is an example. — Thomas Andrews, May 06 '21 at 21:52
$f(x)=f(y)$ doesn’t make a function multivalued. Two different values of $f(1)$ make it multivalued. — Thomas Andrews, May 06 '21 at 21:53

Thomas Andrews · Answer 1 · 2021-05-06T22:57:11.007

Yes, in complex numbers, $\log z$ can be thought of as a multi-valued function. If $w$ is one value, $w+2\pi k i,$ where $k$ is an integer, are the other possible values.

It is abuse of notation to write $a=f(z)$ when $f$ is multivalued, because we lose the meaning of equality in that case: $1=\sqrt 1=-1?$ We still use notation this way, but you have to be careful.

We can think of the multivalied nature of $\sqrt{z}$ as a result of that of l$\log(z).$ We can define:

$$\sqrt{z}=z^{1/2}=e^{\log(z)/2}.$$ While $\log(z)$ has infinitely many values, this definition of $\sqrt z$ only takes two values. (Why?)

In fact, of multivalued analytic functions on $U=\mathbb C\setminus\{0\},$ $\log$ can be considered a father of them all - every other multivalued analytic function on $U$ can be written as $g(\log z)$ for some single-valued analytic $g.$

The very term “branch cut” gives a hint of what we are doing. We are picking a region in the complex plane, and a single-valued function on that region, removing the other values (branches.)

We usually choose branch cuts along lines, but we don’t have to.

The graph of a single-valued analytic function defined on a set $U\subset \mathbb C$ is topologically the same as $U,$ under the obvious map.

But the “graph” of multivalued complex analytic functions like $f(z)=\sqrt z$ and $f(z)=\log z$ on $U=\mathbb C\setminus \{0\}$ are “covering spaces for $U.$” In particular, given any path from $z_1$ to $z_2$ not through $0,$ and one value of $w_1=f(z_1)$ we can find unique values of $f(z)$ along the path which make $f$ continuous along that path, and get a unique value for $w_2=f(z_2).$

In the case $z_1=z_2,$ we don’t in general have $w_1=w_2.$ For example, if the path goes around $0$ once from $z_1$ back to $z_1,$ and $f(z)=\sqrt z,$ then $w_2=-w_1.$

The functions discussed so far are inverse functions of single-valued functions, $e^w$ and $w^2.$ The graph of $e^w$ is the same space as the graph of the multivalued $\log(z),$ with coordinates reversed.

But some multivalued functions are more surprising, like $\sqrt{z(1-z)}.$ This can be made to have a branch cut on the closed real interval $[0,1].$ A path that crosses the interval $(0,1)$ once changes the sign. Twice reverts it.

In the 4th para from the bottom, do you mean $w_2=f(z_2)$ instead of $w_1=f(z_2)$ ? — MAS, May 06 '21 at 22:28

Branch cut of $\log(z)$ in comparison with branch cut of a function like $\sqrt{z}$

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