Talk:Branch point

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Re: Changed Formula

A point about the now-changed formula

F(z)={\sqrt {z}}{\sqrt {1-z}}

is that it is no innocent difference, to take the square root of each factor separately; each then has a different domain of definition, so that F is actually defined a priori on the intersection of the domains. Which is to confuse the point being made with the example, really.

Charles Matthews 11:05, 15 Oct 2004 (UTC)

Mistake

Is the statement:

A point z₀ is a branch point for a holomorphic function f(z) if and only if its derivative f ′(z) has z₀ as a simple pole (i.e., a pole of order 1) − see mathematical singularity.

correct? The function ${\sqrt {z}}$ seems to be a counter example. 0 is a branch point, but its derivative is 1/2 ${\sqrt {z}}$ which also has a branch point at zero, not a pole of order 1.

It's a strange thing to say. Branch points basically are places where the inverse function theorem fails, in this setting. So they are associated with zeroes of the derivative of the function to be inverted. Charles Matthews 09:55, 16 Jan 2005 (UTC)

That's funny, because I found your statement strange. :) I see your point, and I understand what you are trying to say. However, let us be rigurous.

Err?! The derivative of

{\sqrt {z}}

is

1/(2{\sqrt {z}})

which does have a simple pole at zero. —Preceding unsigned comment added by 129.97.218.32 (talk) 03:13, 4 April 2008 (UTC)[reply]

A branch point is not a place where the inverse function theorem fails. For the function z->z² the inverse function theorem fails at 0, but 0 is in no way a branch point for this function. A branch point, according to what the article says, is a point where the function is multi-valued, like log z at 0.

The inverse of z->z², which is the square root, does have 0 as a branch point. However, the derivative of the square root does not have 0 as a pole.

In short, there is some connection between branch points and poles, but the statement

A point z₀ is a branch point for a holomorphic function f(z) if and only if its derivative f ′(z) has z₀ as a simple pole (i.e., a pole of order 1) − see mathematical singularity.

is still wrong. I am talking wrong mathematically, that statement cannot be proved to be right, because I gave a counterexample (the second item above). Oleg Alexandrov 17:23, 16 Jan 2005 (UTC)

You are correct, of course. I was trying to put the ramification concept into quite elementary language. Charles Matthews 19:18, 16 Jan 2005 (UTC)

?

perhaps a definition? —The preceding unsigned comment was added by Rsjyufoih (talk • contribs) .

Confused by Square Root Example

I find the example for the square root to be confusing. The example says that a branch point of sqrt(z) is zero, but the suggested domain ("a circle of radius 4 centered at 0") never passes through zero. The suggested domain seems to be z=4*e^(i*theta). The values of the square root for this domain trace a semicircle of radius 2 centered at 0" -- z=2*e^(i*theta/2) -- which, likewise, never passes through zero.

Elsewhere in the article: "it is customary to construct branch cuts in the complex plane, namely arcs out of branch points". This seems to suggest that the shape of the arc is determined by selecting branch points individually. Would it be accurate to say that a branch point is an intersection of the value and/or domain of the function with a branch cut?

Which would be the branch point(s) in the given example for the square root: "4+0*i" (a point in the domain) and/or "2+0*i" (a value of the function)?

Using the same subscript throughout the following statement suggests that a branch point z₀ is necessarily at the origin: "a branch point may be informally thought of as a point z₀ at which a 'multiple-valued function' changes values when one winds once around z₀." Ac44ck 03:55, 29 May 2007 (UTC)[reply]

There's no "suggested domain" at all. The circle of radius 4 centered at 0 is not suggested to be the domain. Rather, it is used for the purpose of showing why 0 is a branch point. If you go around 0, following this path, the value of the function when you get back to your starting point is no longer what it was when you started. That's why 0 is a branch point. Michael Hardy 21:27, 29 May 2007 (UTC)[reply]

The discussion in another section of this talk page seems to say, "A branch point .. is a point where the function is multi-valued." This suggests to me that there is a branch point _on_ the circle of radius 4. The article mentions a branch point only at zero.

The mention elsewhere in the article that branch cuts are constructed from "arcs out of branch points" suggests that it is common for a function to have several branch points.

If the center of the circle is the branch point in the given example, then such would seem to be the case for a circle of any radius -- and there is only one branch point for the square root function: at zero. Is this what the example is showing?

Ac44ck 04:26, 16 June 2007 (UTC)[reply]

That 0 is a branch point of the square root function is what it is saying. It stops short of explicitly saying "only", although "only" would be correct. Any circle that goes once around the origin will do; the number 4 was chosen for numerical convenience. Michael Hardy 00:10, 23 September 2007 (UTC)[reply]

Mathworld seems to say that there are an infinite number of branch points:

http://mathworld.wolfram.com/SquareRoot.html

"the complex square root function has a branch cut along the negative real axis."

According to the current article, branch cuts are constructed from "arcs out of branch points".

It seems to me that the branch point for the circle of radius 4 occurs where that circle intersects the negative real axis -- not at the origin.

-- Ac44ck 08:18, 23 September 2007 (UTC)[reply]

Some confusion here. A point on a branch cut will be a kind of jump point, but is not what is known in the trade as a "branch point". The branch cut is in any case a somewhat arbitrary line: after following the negative real axis for a bit, it could suddenly head off south-west. This would be an awkward way to define "square root", but would be well-defined. And the points at which the square root function then could not be made continous would be different. A branch point, on the other hand, has a meaning intrinsic to the squaring function we are trying to invert. The point 0 is a branch point because of the geometry there of squaring. Charles Matthews 09:58, 23 September 2007 (UTC)[reply]

What does this mean?

If you can explain what the following means, then probably the section is alright:

A point z in the domain of f is said to be a branch point of f if the value of f at z differs depending on its argument. That is, z is a branch point of f if and only if for two particular arguments θ₁ and θ₂ of z, the value of f at z is different for each.

However, please be aware that a holomorphic function of the complex variable z must necessarily have the same value whether one chooses θ or θ+2π for the argument of z. So until this matter is cleared up, the definition is unclear and non-rigorous. I suggest that it be removed, or at least clarified. In the present state, it is utterly nonsensical. siℓℓy rabbit (talk) 20:58, 10 December 2008 (UTC)[reply]

Copied from silly rabbit's talk page:

OK. I will remove the first definition. The second definition probably explains it in a more formal manner.

Topology Expert (talk) 21:13, 10 December 2008 (UTC)[reply]

Better? I removed everything that is disputed and kept what is not. When we sort the issue out, we can add back what is correct. I have to sleep now so I will respond to comments tomorrow. If you still think that there are problems with this definition, please explain them. Tomorrow I will clarify definition 1 and 'holomorphic' (and anymore issues).

Topology Expert (talk) 21:18, 10 December 2008 (UTC)[reply]

Good night.

Topology Expert (talk) 21:18, 10 December 2008 (UTC)[reply]

The definition seems alright but I will have a proper look at it now.

Topology Expert (talk) 09:29, 11 December 2008 (UTC)[reply]

Some questions

Just a few questions. The formal definition section associates branch points with the original function, but the examples section associates the branch points with the inverse maps. There is a note explaining this in the formal definition section (which is helpful), but we should probably be consistent in how we introduce the branch points, do you think? For example, in the first example we could introduce the function ƒ:C -> C by ƒ(x) = x^2, deduce that 0 is a branch point, and leave some notes about the square root function. Maybe we should extend this example to the Riemann sphere, since this could easily double up as a Riemann surfaces example then too, and note that $\infty$ is a branch point too. I'm happy to do this if you all agree, but this brings me to my next question.

In example 2, we can't deduce that 0 is a branch point of the logarithm function using the method explained in the formal definition section, since the logarithm function isn't even defined at 0. So I'm wondering if we should modify the formal definition section a bit too? Maybe even have a complex analysis section replace this section? Then we can have examples fleshed out from each of complex analysis, Riemann surfaces and algebraic geometry, though I can't help with the last two.

In Forster's 'Lectures on Riemann Surfaces' (GTM 81), he gives the definition of a branch point as follows: Suppose X, Y are Riemann surfaces, p : X -> Y a non-constant holomorphic map. A point y in Y is called a branch point or ramification point of p, if there is no neighbourhood V of y such that p|V is injective. Looking at the Riemann surfaces section of this article and the local coordinates article, I see some similarities, but in particular, Forster doesn't seem to differentiate between ramification points and branch points. This is a bit different to our formal definition too, so I'm wondering if maybe this Forster definition isn't so widepsread? Or if it is, maybe this should be noted at some point? Cheers, Ben (talk) 12:27, 11 December 2008 (UTC)[reply]

Hi Ben. You make some excellent points and I must say I'm sorry that I apparently missed this thread in all of the hullabaloo. Unheeded, I had some of these things in mind while editing this evening, and so I have probably at least partially cleared up some of the ambiguities here (although some confusion still seems to remain: see the next thread). If it isn't too much to ask, perhaps you would care to examine the article once more with the same critical eye?

With regard to your last point, the Forster definition is equivalent to the one in the Riemann surfaces section. It is somewhat simpler however, and so probably should be included in the article, and stated before the more sophisticated definition involving the ramification index. The distinction between branch points and ramification points is a fairly common one, although not universal. siℓℓy rabbit (talk) 01:56, 12 December 2008 (UTC)[reply]

Logarithm

The singularity of the logarithm does not qualify as a branch point according to most definitions. It is not in the domain of the function, and doesn't come from any good ramified cover. It is of a special enough kind that it deserves some treatment, but separately from the main cases. However, most of the examples in the article are logarithmic branch points. I think this should be fixed. siℓℓy rabbit (talk) 12:44, 11 December 2008 (UTC)[reply]

So what do you call a thing like that if not a branch point? Are you saying something is not a branch point unless it's in the domain, i.e. it gets mapped to some finite complex number? Michael Hardy (talk) 13:15, 11 December 2008 (UTC)[reply]

Domain also generally includes poles. The term "branch point" typically means what some would call "algebraic singularity" (see for instance Ahlfors). I have heard the singularity of the logarithm referred to as a "logarithmic branch point". It is, of course, an essential singularity rather than an algebraic pole. siℓℓy rabbit (talk) 13:26, 11 December 2008 (UTC)[reply]

I don't agree with this--- the log singularity is only special because you get infinitely many values when you go around. With square root branch cuts you get two values, but

{\sqrt {x}}=e^{\log(x)/2}

and exp is analytic. So the square-root branch cut (the "joe sixpack" of branch-cuts) is exactly the same as the log branch cut after some analytic gluing.Likebox (talk) 19:13, 11 December 2008 (UTC)[reply]

But exp is a covering map, not a ramified cover, so there is no branch point in the sense that is usually meant. Of course, there is monodromy (and so one can talk about branch cuts), but branch cuts are a different thing. siℓℓy rabbit (talk) 19:25, 11 December 2008 (UTC)[reply]

The only difference between the two is that there are infinitely many values when you go around the log "branch point". The rational exponent branch points have a finite number of values, so they are special--- they can be "unwound" with a rational map. The log branch point requires a transcendental map which has an essential singularity at infinity. This is a difference, but it is not so important a difference in my opinion to declare that "log" doesn't have a branch cut.

If you separate the two cases, then you would also make a huge distinction between x^\alpha for \alpha rational and \alpha irrational. In the rational case, you have a ramified cover, and in the irrational case, you have the same log-singularity. When people say "branch cut" in the physics literature, they mean a continuum of poles, and the two cases are never separated--- they are both "cuts".

For purposes of algebraic geometry, I agree that the two cases should be separated. The right way, I think, is to give names to the two cases: finitely ramified branch cuts and logarithmic branch-cuts. The separation is like that between rational and irrational numbers, for certain idealized properties, the two are very differnt, but for applied mathematics the two continuously blend together.Likebox (talk) 23:06, 11 December 2008 (UTC)[reply]

I think we may actually be in agreement, then. I am not saying that logarithms do not belong here, but that they should be kept separate from the algebraic case, and that there is no need to try to give a general definition which will simultaneously address all concerns. I think the definition for algebraic singularities is the simplest, and should go first. Transcendental singularities seem to require the machinery of analytic continuation, and so should be second. This also gives the article some time to motivate the introduction of analytic continuation and global analytic functions to begin with. siℓℓy rabbit (talk) 23:37, 11 December 2008 (UTC)[reply]

(deindent) Root branch cuts also need analytic continuation, and I personally think logs are simpler. But whatever floats your boat. I think the current wording is decent.Likebox (talk) 23:54, 11 December 2008 (UTC)[reply]

Actually, algebraic branch points only require the notion of a covering: a branch point is the image of a ramification point. Analytic continuation is not required. The first section of the article, after the lead, defines a branch point without any reference at all to analytic continuation. siℓℓy rabbit (talk) 01:13, 12 December 2008 (UTC)[reply]

And how do you define the function on the cover? By analytic continuation from some known values. It's exactly the same in the case of the log function. You can define the Riemann surface of a "spiral staircase", and a projection onto the complex plane, and the log is single valued. There's really no difference of principle, the only difference is that you can't straighten out the singularity algebraically.Likebox (talk) 02:01, 12 December 2008 (UTC)[reply]

I agree that one needs analytic continuation to define the function via branch cuts. However, note that I am here only talking about the branch points. (Isn't that what the article is supposed to be about?) While a complete description of constructing the analytic continuation of a function in terms of its function elements is probably beyond the scope of this article, a section should probably be added somewhere on how to define analytic continuations via branch cuts. siℓℓy rabbit (talk) 02:16, 12 December 2008 (UTC)[reply]

I see. Thanks.Likebox (talk) 02:20, 12 December 2008 (UTC)[reply]

Holomorphic at the branch point?

The article now says:

Algebraic branch points most commonly arise from functions in which there is an ambiguity in the extraction of a root, such as solving the equation z = w² for w as a function of z. Here the branch point is the origin, because the analytic continuation of any solution around a closed loop containing the origin will result in a different function: there is non-trivial monodromy. Despite the algebraic branch point, the function w is defined and, in an appropriate sense, is holomorphic at the origin.

This seems to say that the square root function is holomorphic at the origin. In what sense is that true? Michael Hardy (talk) 23:25, 11 December 2008 (UTC)[reply]

I mean that the branched covering map z=w² is defined and holomorphic over the origin as well, which is the chief feature distinguishing between the algebraic and non-algebraic cases. I'll give it another go tomorrow. siℓℓy rabbit (talk) 00:59, 12 December 2008 (UTC)[reply]

Expert attention: branch point $z_{0}$ or $f(z_{0})$ ?

There seems to be an inconsistency in the Algebraich branch points section. In the second paragraph $z_{0}$ is said to be a ramification point and $f(z_{0})$ is said to be a branch point, whereas the third paragraph speaks of "a branch point $z_{0}$ of $f$ ". This may or may not be related to the abuse of language mentioned in the third paragraph. If it is, it needs to be made much clearer which function we are talking about at which point, and ideally different symbols should be used for points in the domain of $f$ and points in the domain of $f^{-1}$ . Joriki (talk) 07:37, 10 April 2009 (UTC)[reply]

An anonymous user has since made some helpful improvements, but problems remain, so I've added the expert attention template again (this time for the article, since the problems are partly in the relationship between the sections). Some remaining problems are:

"Then g is said to be a transcendental branch point" makes no sense (g is a function) -- this should probably be either $z_{0}$ or $g(z_{0})$ .
The use of the term "branch point" still seems inconsistent between the sections "Algebraic branch points" and "Transcendental and logarithmic branch points". It's one thing to note that often no distinction is made between branch points and ramification points in the literature, and another to use the terminology introduced in the article within the article with a different meaning than was introduced.

Joriki (talk) 12:02, 19 April 2009 (UTC)[reply]

The use is consistent between the two sections. One speaks of branch points of the square root, for instance, and ramification points of the squaring function. I have also removed the "expert" tag and corrected the grammatical lapse that you objected to in your first bullet point. 71.182.187.38 (talk) 11:26, 24 April 2009 (UTC)[reply]

Branch points in fields?

I'm a bit confused about the lead section of this article; to me it seems to indicate that branch points is something that only exist in complex functions. But what about scalar or vector fields? Don't branch points exist in those too? —Kri (talk) 10:38, 28 August 2016 (UTC)[reply]

Definitions are wrong, or at best imprecise

The definition in the lead makes no sense. One can say that a single-valued function is discontinuous at a point (i.e., not continuous there), but what does it mean to say that a multi-valued function is "discontinuous when going around a circuit"? Also, the definition for Riemann surfaces given later is too limited since it requires both $X$ and $Y$ to be compact. Better references need to be found. Ebony Jackson (talk) 18:15, 23 January 2021 (UTC)[reply]

just changed the the definition in the lead you've mentioned. this one might still be slightly imprecise and could use an expert's second opinion but it I think it's at the very least using well defined statements and much less imprecise partly because of it. Axiomatic Ginger (talk) 20:41, 27 April 2022 (UTC)[reply]