Talk:Zero (complex analysis)

Suggestion

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)

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 $g(a)\ne 0$ AND $g(a)\ne \pm \mathrm{\infty }$? Monsterman222 (talk) 20:17, 13 December 2011 (UTC)

naming convention

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)

Clarification: Isolated zeros only if single variable

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)

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Zeroes isolation proof?

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)