Talk:Branch point

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[edit] 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)

[edit] 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)
  • 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)

[edit] ?

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

[edit] 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)

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)
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)

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)

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)
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)