Talk:Zero (complex analysis)

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The content of the Zero (complex analysis)Zero (complex analysis) page was merged into Zeros and poles, and it now redirects there. For the contribution history and old versions of the merged article, please see its history.

What is said about entire functions isn't right; no zeroes is equivalent to having a well-defined logarithm without branch points, but any function exp(f(z)) with f entire will do.

This article should surely mention the contour integral way to count zeroes (integrate the logarithmic derivative); Rouché's theorem; and perhaps the construction of functions with given zeroes (infinite products, Blaschke products).

Charles Matthews 16:27, 10 Jun 2004 (UTC)

I think this page should be merged with root(mathematics) and a reference to rouché' theorem should be given. other opinions? Hottiger 15:28, 12 April 2006 (UTC)Reply

I like it more this way. If merged, the new root (mathematics) will be unnecessarily biased towards complex analysis in my view. Other views? 23:26, 12 April 2006 (UTC)

Shouldn't the fourth line of the Section Multiplicity of a Zero read: "Generally, the multiplicity of the zero of f at a is the LEAST positive integer n...", or if you don't want to say that, then wouldn't we need to specify that ${\displaystyle g(a)\neq 0}$ AND ${\displaystyle g(a)\neq \pm \infty }$? Monsterman222 (talk) 20:17, 13 December 2011 (UTC)Reply

I think it would be helpful, to -at least- add the notion of "root" instead of "zero".

The "zero" of a function should simply be its value.

In other articles in wikipedia, the value of x, where a function f(x) of x has its value f(x)=0, is called a "root" of the function.

It would be helpful, to -at least- introduce the crossreference (term "root") here.

--Gotti 10:15, 12 March 2007 (UTC)

I added "in one variable" to the sentence "An important property of the set of zeros of a holomorphic function (that is not identically zero) is that the zeros are isolated" because, for example, the two-variable holomorphic function

f(x,y)=xy-1

is zero in the circle a^2 + b^2 = 1, where x=a+ib and y=a-ib. —GraemeMcRae^talk 16:00, 15 February 2010 (UTC)Reply

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The "Properties" section of the article as of now states "An important property [...] is that the zeros are isolated" without providing any justifications. No matter how trivial, wouldn't it be fair to include a proof of this statement?

(Sketch of a proof: non isolated zeroes => exists sequence of zeroes with adherent point => exists converging subsequence => recursively all nth derivatives of f are null at point of convergence of the subsequence => f = sum nth derivative / n! etc. = 0 on the connected domain containing the point of convergence on which f is holomorphic i.e. coincides with its series)

I understand this proof may need formatting, I lack the confidence of brute force inserting it into the article. 46.193.1.224 (talk) 08:29, 12 March 2017 (UTC)Reply