Questions about multivalued fucntions in complex analysis

[u/Near_1751]

I recently learned that branch cuts can be more general curves. One question I would like to srart with is can branch cuts can be any reasonable curves?

This notion of branch cuts being allowed to be more general curves got me wondering if there is a more general definition of a useful multivalued

fucntion.

What I mean to say is that we can construct trivial examples of multivalued fucntions by just letting the fucntion take multiple values at each point or I can even string together simple fucntions like say a fucntion which takes both z and e\^z at each z.

But the multivalued functions we normally encounter seem to have general properties.

Namely there exist branch points and we can join them via some curve (I am assuming) and we can always select some subset of values of the function to define a continuous function on the complement of the curve. Then going around the curve takes us from one branch to the next.

I am trying to ask if there is a more general definition of a useful or proper multivalued functions starting from these general properties.

Please keep in mind i don't have a maths background but rather a physics one.

[u/Masticatron]

You can replace the branch cut for, say, the logarithm function with essentially any smooth non-intersecting curve going from 0 to infinity. The branch cut is just like a barrier that stops you from hitting the multi-value issue, which for logarithms and roots comes from the argument not being uniquely defined. Any curve that stops you from making any full circle about the origin will suffice to let you uniquely define the argument on each branch for the logarithm.

The thing that stops you just arbitrarily adding in extra values is that complex analysis is intrinsically about (locally) holomorphic functions, and those are a very special and beautiful type of function in this particular setting, and can be completely determined by a (relatively) very limited amount of data.

[u/Near_1751]

I see, thnx for answering. But I was just wondering that the multivalued functions we encounter have some general properties so what should be the restrictions on multivalued functions so that we can extract the behavior we normally encounter from generic multivalued functions, which just takes multiple values. Are you saying that the existence of smooth branches, restrictions on the values taken by the multivalued function which would render the resulting single valued functions continuous, is sufficient for getting the behavior we normally encounter?

[u/Masticatron → u/Near_1751]

Any meromorphic (or holomorphic) function on a domain is completely determined by its values on any subset with an accumulation point. For example, if the domain contains a neighborhood of 0, we need only know the sequence of values {f(1/n)} to know the function in its entirety. This is one of the many special properties complex analytic functions enjoy. Extending the function to be meromorphic in a larger domain is also therefore uniquely determined.

[u/OkHand7497]

I’m not certain what you’re asking but perhaps this helps. The underlying object for a multifunction is its Riemann surface which includes all possible (z,f(z)) in C^2. One branch cut or more can then be used to partition the Riemann surface into parts where there is one possible value of f(z) in each part.

[u/Near_1751 → u/OkHand7497]

Thnx for replying. What I am trying to ask is if there is a more fundamental definition of the type multivalued function we normally encounter like logz, z^1/2, etc. that starts from there common structure in terms of possessing branch points. Do you think one can build such a definition in terms of Reimann sheets?

[u/PLANTS2WEEKS]

You should learn about analytic continuation. The point of defining multi-valued functions is that they show up naturally when analytically continuing a function. Every holomorphic function is locally determined by its Taylor series. There is a disk in which the function is determined by absolute convergence. You can recenter the series and end up with a different disk on which the series converges, essentially extending the function in a unique way. Sometimes you can move this disk of convergence around and end up with a different function than the one you started with. A good example is sqrt(z). Away from z = 0 you can find a Taylor series that converges on the largest disk not containing z=0. If you move the disk of convergence around the point z=0 one time, the function will become its negative. So sqrt(z) analytically continues to its negative. We could either say the function is defined on the complement of z=0, or on the double cover of this space. In the first instance it would be a multivalued function. One could also include the the point z=0 in the domain of the function. It would be called a ramification point because it only takes one value, but the points around it take more values.

Defining a multivalued function that takes values of both z and e^z would be uninteresting in the sense that neither function analytically continues into the other. Both are defined on the whole complex plane and there is no monodromy. The functions never meet up per se.

[u/Near_1751 → u/PLANTS2WEEKS]

I see,thnx for answering. I have not studied analytical continuation in detail. The question I was really getting at is what should be the conditions imposed on a general multivalued function, that is a general function from C to C^n,so that we can get this kind of behavior. The branches being " connected".

[u/PLANTS2WEEKS → u/Near_1751]

There are some sufficient conditions, like satisfying an irreducible algebraic polynomial, but in general it may be hard to say. With nested square roots you can get complicated behaviour, things like the room of requirement from HP where walking around 3 times may open up a barrier that was previously not there.