How to find principal branch of a complex multivalued function?

Suppose I have a complex multivalued function $$\log(f(z))$$, and I am required to find the principal branch of this function.

The method I have learned says that the principal branch of $$\log(z)$$ is obtained by restricting the argument from $$-\pi$$ to $$\pi$$. As a consequence, the branch cut is the negative real part along with the origin. Using similar logic for $$\log(f(z))$$ we get the principal branch with the branch cut $$\Re(z) < 0$$ union $$(\Re(z) = 0, \Im(z) = 0)$$.

My main problem with this method is: How does this ensure that the branch obtained is the principal branch? I can restrict the argument from $$-3\pi$$ to $$-\pi$$, $$\pi$$ to $$3\pi$$, etc all which will satisfy $$\Re(z) < 0$$ union $$(\Re(z) = 0, \Im(z) = 0)$$.

• Do you have a definition of "principal" that you want to prove this branch satisfies? As far as I know "restrict the argument to $(-\pi,\pi]$" is the definition of the principal branch of the logarithm, but there is no general definition of the word that applies uniformly to all multivalued functions. Jun 21, 2021 at 10:53
• Yes, that is what I meant by principal branch. Jun 21, 2021 at 10:54
• Perhaps it would be helpful if the question explained exactly why you need the principal branch. If it is an exercise from a book, for example, you could include the author, title, chapter, and problem number (or other identification) and the exact text of the problem. Jun 21, 2021 at 11:38

1 Answer

The pesky thing about multi-valued functions is that there is no general way of choosing a branch cut that will apply to all functions in a uniform way. The words "principal branch" don't mean anything until you have chosen a definition for the particular function you're looking at, and it doesn't make sense to ask whether the definition you choose is "correct" (as long as it does select some branch, of course). The definition you choose is the definition you choose.

The logarithm is one of the very few multi-valued functions where a common definition of "principal" appears in textbooks, which is nearly always one of

1. The principal branch of the logarithm is the one where you always select the argument to be in $$(-\pi,\pi]$$.

2. The principal branch of $$\log(z)$$ is the one created by always taking the function value whose imaginary part is in $$(-\pi,\pi]$$.

These definitions happen to be equivalent for the logarithm function -- but if you take a general multi-valued function and attempt to apply these definitions, one or both may fail to make sense at all -- or they may lead to different results.

For example, imagine that you have a random power series with positive radius of convergence, and someone says "now take the maximal analytic extension of this". In general you get a multi-valued function, but there's no choice of argument in view anywhere, so the first definition doesn't make sense. And there's definitely no reason to expect that among the possible values of the function at some point exactly one will have imaginary part in $$(-\pi,\pi]$$.

When you have a function defined by the expression $$\log(f(z))$$ and $$f$$ is single valued, it will often make sense to decide to consider the branch defined by $$\operatorname{Log}(f(z))$$ to be "principal" -- which in this particular case corresponds to either of the definitions above. In that case, however, you have nothing to prove: You've just made a choice, and however excellent that choice is, there's no generally applicable nice property you can prove it has.

This way of choosing doesn't even always behave nicely when you do have explicit expressions for the function. For example the expressions $$z\mapsto \pi i + \log z \qquad\text{and}\qquad z \mapsto -\pi i + \log z$$ define the same multi-valued function, but if you blindly try to apply definition 1 above to the expressions you get two different choices for a principal branch out of them. And if instead you apply definition 2 (which just happens to make sense in this case too) you get a third, still internally consistent, choice.

Finally, the wording in your question sounds like you're expecting that if only you know the location of the branch cut you can also know which branch you're talking about in the rest of the complex plane. That is definitely not true. The function values are what matter for the branch. The location of the cut is derived from your choices of function values, not the other way around.

Consider that the branch cuts are what separate the branches in the Riemann surface, so a particular cut will also appear as the boundary of a neighboring branch (or more, as in the logarithm where you get to different branches if you move past the cut clockwise or counterclockwise).

(Of particular relevance to your question: if you have for example $$g(z)=\log(\sin(z))$$, then interpreting the expression by using the principal value of the logarithm will give you a branch choice whose cuts look very different from "the negative real axis" -- there are no branches of $$g$$ that cut only at the negative reals).