# Do we need a branch cut to choose a branch? If we arbitarily choose the initial value $$w=\ln(2 \sqrt{2}) + i \frac{\pi}{4}$$ for $$\ln(2 +2i)$$ , then as $$z$$ travels along a loop the encircles the origin $$\nu$$ times, $$\log(z)$$ moves along a path from $$\omega$$ to $$\omega + 2 \nu \pi i$$. Check that you understand (roughly the shapes of the illustrated paths

Pg-98,99, Tristan Needham Visual Complex Analysis

1. How can he speak of a multi-valued function as a single valued one without a branch-cut?

2. If chooses a branch for evaluating the multi-valued function as a single valued function, would that implicitly put a branch cut? I think it does, and then, I am confused because how he was able to loop back along the dark circle back to $$2 + 2i$$ because the points on the positive $$x$$-axis are removed.

Because if he would have done a branch cut while choosing a branch, then why can he move around this curve in the picture, I mean then there would be a branch cut so a "line" where he can't go further since otherwise he would get to the multivalued issue again.

Obviously $$\log(z)$$ is a multivalued function as explained on p.98. To define single-valued branches of $$\log(z)$$ the author introduces a branch cut from $$0$$ to $$\infty$$ and says that the most popular choice is the negative real axis (see p.99). This should answer your first question: The author does not speak of a multi-valued function as a single-valued one without a branch-cut.

After having made the branch cut we can define the principal branch of the logarithm based on the principal value of $$\operatorname{Arg}(z)$$. This is a function

$$\operatorname{Log} :\mathbb C \setminus \{0\} \to \mathbb C .$$

The problem is that $$\operatorname{Log}$$ is not continuous at the points of the branch cut. You can still move along any closed curve $$u : [0,1] \to \mathbb C \setminus \{0\}$$, and always end with the same value of $$\operatorname{Log}(z)$$. But this is not a valuable information, $$\operatorname{Log} \circ \ u : [0,1] \to \mathbb C$$ is not continuous if $$u$$ crosses the branch cut.

Unfortunately Needham does not say that he wants to get continuous branches. Anyway, one can easily show that there does not exist a continuous $$\operatorname{Log} :\mathbb C \setminus \{0\} \to \mathbb C$$. This can be resolved by removing the branch cut $$B$$ which gives a continuous $$\operatorname{Log} :\mathbb C \setminus B \to \mathbb C .$$ In this situation he is in fact no longer able to loop back along the two circles in Figure  back to $$2+2i$$ because the points of the negative $$x$$-axis are removed. But be aware that his original $$\operatorname{Log}$$ is defined on $$\mathbb C \setminus \{0\}$$.

To define a "single-valued branch" of the logarithm without any further assumpions on $$\operatorname{Log}$$ we can choose for each $$z \in \mathbb C \setminus \{0\}$$ an arbitrary element $$\operatorname{Log}(z) \in e^{-1}(z)$$. The resulting function will in general be highly discontinuous. The principal branch is a special choice which is as continuous as possible; the discontinuities lie on the branch cut.

• First of all thank you for your answer and sorry that I respond so late. Did I understand it correctly that in his example the function is not continuous? So I mean only when we remove a branch cut we get a continuous single valued function which we can work with. But removing this branch cut also leads to the "problem" that we can't turn around the origin. Jun 13, 2022 at 11:23
• @Wave Yes, this is correct. Jun 13, 2022 at 12:30
• Ah but then even if it's not continuous he can move along it. Jun 13, 2022 at 12:57
• But for computations we always need to remove a branch cut right? Jun 13, 2022 at 12:57
• @Wave It depends on what you want to do. If you just want to assoiciate to $z$ a single value of $\log z$ and do not care about continuity, then you do not need to exclude the branch cut. But if continuity is essential in some context, you must remove it. As a branch cut we can choose any ray starting at $0$ an going to $\infty$. At the points of the branch cut the $arg$-function jumps by a value of $2\pi$ if you travel along a circle. Jun 13, 2022 at 13:04