Talk:Branch point

message: 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)
thread_title: Re: Changed Formula

message: 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)
thread_title:  What does this mean? 

message: 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)
replies: text: 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)
thread_title:  Holomorphic '''at''' the branch point? 

message: 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)
thread_title:  Branch points in fields? 

message: 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  and  to be compact.  Better references need to be found. Ebony Jackson (talk) 18:15, 23 January 2021 (UTC)
replies: text: 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)
thread_title:  Definitions are wrong, or at best imprecise