Talk:Analytic continuation

message: Ughh. Someone should copy the bottom half of the article on Riemann surfaces into this article.  The article on Riemann surfaces should be trimmed to remove the parts about analytic continuation, and just point to this article instead. Who is up for that?  linas 19:43, 22 Jan 2005 (UTC)
replies: text: There also should be a mentioning of the generalization to meromorphic continuation since the link points to this article currently. - Gauge 05:16, 20 May 2005 (UTC)
thread_title:  Trim 

message: The formula
:L(z-1) = \sum_{k=1}^\infin \frac{(-1)^{k+1}}{k}(z-1)^k
is weird. Perhaps the author wanted to write
:L(z) = \sum_{k=1}^\infin \frac{(-1)^{k+1}}{k}(z-1)^k
Bo Jacoby 13:47, 10 October 2005 (UTC)
replies: text: OTOH perhaps the author meant log(z) = L(z-1) = ... and "L(z-1)" is only there to emphasize the fact that z0=1. 194.78.219.157 18:28, 10 October 2005 (UTC)

text: And its not really an example of analytic continuation is it (its a power series with a statement of what its germ is)? Its a concept that I am just trying to understand and there are surprisingly few examples of it being done. Instead one reads lots of earnest accounts of how an analytic continutation (if it exists) is unique in certain circumstances and how easy it is to do (or is it). Nowhere is it said (as far as I can see) under what conditions such a thing exists or how to find one (although its implied that this is so easy it should be obvious). Fine for a typical maths textbook, but not (surely) an encylopedia. How about an *example*? Francis Davey 19:02, 11 May 2006 (UTC)
thread_title: Examples of analytic continuation

message: The reference to "global obstructions" is not immediately clear to me. The article on "Sheaf Cohomology" doesn't help about. Better specify. 9:05, 20 May 2006
:I've written this out in a longer explanation. Charles Matthews 11:17, 20 May 2006 (UTC)
thread_title:  Global obstructions 

message: "That is because the difference is an analytic function vanishing on a non-empty open set."
This needs further comments, because i don't get it.
:I gave it a try.  How is it now? -lethe talk [ +] 16:12, 11 June 2006 (UTC)
replies: text: Not "approaches zero", but rather "is equal to zero at every point of the domain". -lethe talk [ +] 06:14, 12 June 2006 (UTC)
thread_title:  this sentence 

message: The article states:
"Sum_{1 <= k < oo}  (-1)^(k+1) (z-1)^k is a power series corresponding to the natural logarithm near z = 1. This power series can be turned into a germ g = (1, 0, 1, −1, 1, −1, 1, −1, ...)."
Shouldn't the germ be g = (1, 0, 1, -1/2, 1/3, -1/4, ...) (using the notation for germs described in the paragraph above this statement) ???Daqu (talk) 09:57, 23 May 2008 (UTC)
replies: text: agreed. done. Bo Jacoby (talk) 12:27, 23 May 2008 (UTC)
thread_title: The example of log(z) seems wrong

message: This sentence: "The base g0 of g is z0, the stem of g is (α0, α1, α2, ...) and the top g1 of g is α0. The top of g is the value of f at z0, the bottom of g."  I couldn't find this terminology by a Google search. Also, z0 is referred to as both the 'base' and the 'bottom' of g. Lateralrust (talk) 21:36, 9 March 2011 (UTC)
thread_title: Confusing sentence: Parts of a germ

message: "This is because F1 − F2 is an analytic function which vanishes on the open, connected domain U of f and hence must vanish on its entire domain. This follows directly from the identity theorem for holomorphic functions."
Is this correct? F1 - F2? It seems that would be zero, which is only an analytic continuation of 0. Was this meant to imply something more along the lines of F1 "slash? F2? Using "and" seems like it would be better and avoid extremely likely confusion. 67.194.8.120 (talk) 00:54, 28 March 2011 (UTC)
thread_title:  Typo? 

message: Is there some chance that someone who has both an understanding of the mathematics and a reasonable ability to write comprehensibly might take a pass over this article? I have a B.A. in mathematics, an M.S. in computer science, plenty of theoretical background in mathematical analysis and other related fields, decades of experience as a software developer, etc., and I can't make head or tail of this. If I can't read it, what person who is not already solid on the mathematics in question is going to be able to do so? - Jmabel | Talk 06:47, 29 January 2014 (UTC)
replies: text: Well said!  And I second your request, Jmabel.  I, too, have a similar background.  Yet we're diving into abstract sheaves before we've even bothered to explain the general notion of what we want to achieve or how to go about it.  Just because we may love pure mathematics doesn't mean we're not interested in the concrete details of solving specific problems ... yoyo (talk) 11:32, 5 May 2017 (UTC)
thread_title:  Writing 

message: The picture purporting to illustrate analytic continuation, having the caption "Analytic continuation of natural logarithm (imaginary part)", is misleading.  
In analytic continuation, the center of each disk in a sequence of function elements must lie inside the previous disk.  In this picture, not even one successive disk's center lies within the previous disk.
This is not helpful for illustrating analytic continuation.  Can someone please replace this picture with one that correctly depicts each successive disk as having its center inside the previous disk? (It would be a good idea to display a small dot at each disk's center.)Daqu (talk) 16:34, 12 January 2015 (UTC)
replies: text: Good point, Daqu!  I'm  willing to try improving it, but need some guidance.  I've examined the picture file (on Wikimedia Commons), but that doesn't give any indication of the tools used for creating it, so we'd have to start from scratch (or try to persuade the original creator to improve the graphic.  That was 13 years ago, and the user mostly contributes in Japanese, so good luck if you want to try that way.)  Any recommendations for suitable graphic tools for illustrating complex analysis?  (I mostly use photographic and paint programs.)  And presumably, a sequence of disks each of constant radius R, with centres separated by some large fraction of R (e.g. 99%), would meet the requirements?  yoyo (talk) 11:47, 5 May 2017 (UTC)
thread_title:  Misleading illustration 

message: The description of analytic continuation in terms of germs and sheaves is completely inappropriate as the first definition that the reader encounters.  It would be entirely appropriate if these topics are placed toward the end of the article, as the "modern viewpoint" of analytic continuation. But they are unnecessarily abstract for the first definition that a reader encounters.
The first definition that a reader encounters ought to be simply about a sequence of function elements (power series converging on an open set) such that the center of each successive function element lies within the disk of convergence of the previous one, and such that each successive function element agrees with the previous one where their domains overlap.  This may be thought of as continuation along the curve consisting of the polygon connecting successive function elements' centers with straight lines.
Then, a topology for the function elements should be defined by declaring that continuation along two polygons with the same endpoints is equivalent when the polygons are homotopic to each other, holding the endpoints fixed, within an open region of analyticity.  This (almost) defines the topology of the function (its Riemann surface).  For more information on this, the reader should be referred to the article on Riemann surfaces.
Also: All the stuff about natural boundaries should have its own article.  The concept of "natural boundary" is entirely appropriate for this article, but the theorems about which power series have a natural boundary are too removed from the subject of analytic continuation.Daqu (talk) 19:45, 12 January 2015 (UTC)
thread_title:  Article becomes inappropriately abstract way too soon 

message: The current article states that "The circle is a natural boundary if all its points are singular." But isn't the correct definition "The circle is a natural boundary if the singular points are dense in the circle."?  In my experience, the normal case is that there are an infinite but  countable number of singular points that are dense on the circle. It is because they are dense, that one cannot analytically continue past the natural boundary. I also believe that its impossible to have an uncountable number of singular points on the natural boundary; if one did, the function inside the circle wouldn't be analytic. So an uncountable number of singular points on the boundary is "crazy talk", from what I can tell.  Finally, I think that it is important to note that, if one removes the singular points from the boundary, what remains is a cantor set, viz, the function is regular on a cantor set. One still cannot analytically continue past this, since each "element" of the cantor set is "infinitely small"; the radius of convergence is zero at each such regular point.  I am 99.9% certain that this is the case, but am not aware of which theorems or books convert this into a more precise and accurate statement. (I'm thinking that maybe books on Hardy spaces make such statements, make this more precise.) 67.198.37.16 (talk) 22:20, 10 May 2018 (UTC)
thread_title:  Natural boundary 

message: In the section titled "Formal definition of a germ" the following appears:
"... it would be equivalent to begin with an analytic function defined on some small open set."
What is meant by "small" in this context? 
I suggest "small" be either removed or replaced by some specific quantifier.
Dratman (talk) 08:37, 18 March 2019 (UTC)
replies: text: As far as I perceive the ubiquitous use of the epitheton small ornating open intervals, open balls, open neighborhoods, ..., it is always for allowing these domains to be arbitrarily small in some metric (\varepsilon > 0), just for being able to keep undesired behavior outside the considered range, and since demanding the existence of something that may be arbitrarily small sounds a bit restrictive, the more laissez faire formulation of just small took over. As I perceive it, just removing it may go a bit too far. Purgy (talk) 11:28, 18 March 2019 (UTC)
thread_title:  Small open set? 

message: I can follow it up to the last step, but the last integral just seems to vanish. I assume it is some kind of trick, does anyone know how to do it?
Darcourse (talk) 14:05, 28 April 2022 (UTC)
thread_title:  Last step in 'Worked Example' 

message: After the shift to the new center, the result is:
f(z) = \frac{1}{a-z}
Shouldn't it be just
f(z) = \frac{1}{z}
Otherwise, the original function and the shifted function are not equal in the overlapping region. 2A01:C22:D416:8200:58E8:6611:CDBE:5215 (talk) 19:47, 12 April 2023 (UTC)
replies: text: I agree, so does WolframAlpha: https://www.wolframalpha.com/input?i=1%2Fa*sum+from+k%3D0+to+infinity+%281-z%2Fa%29%5Ek 2001:4CA0:250D:0:5914:C38E:67DB:9480 (talk) 16:15, 13 June 2023 (UTC)
thread_title:  worked example error? 

message: Hello, could we add few steps in one demonstration because I am not understanding two things in those three lines:
\begin{align}
a_k &= \textrm{...}\\
&= \frac{1}{2\pi} \sum_{n=0}^\infty (-1)^n \int_0^{2\pi} \binom{n}{k} (a-1)^{n-k} d\theta \\
&=\sum_{n=0}^\infty (-1)^n \binom{n}{k} (a-1)^{n-k} \\
&=(-1)^k a^{-k-1}
\end{align}
So apparently for all m \neq k the integration is zero because of the loop provided by e^{ik(m-\theta)}. Ok, but then summing over n in \binom{n}{k} creates terms that can't exist if n<k. So should we not change \sum_{n=0}^\infty to \sum_{n=k}^\infty ?
 
In addition, \sum_{n=0}^\infty (-1)^n \binom{n}{k} (a-1)^{n-k} = (-1)^k a^{-k-1} is certainly not the Newton binomial formula, it might be related, but I'm not seeing it.Klinfran (talk) 12:38, 16 February 2024 (UTC)
replies: text: Traditionally, the binomial coefficient is zero if n<k. Nevertheless, an easy change of summation index avoids the use of this convention. The last relation does not results from the binomial coefficient, but from the derivation of the geometric series. I have edited the article for making the article clearer. D.Lazard (talk) 15:44, 16 February 2024 (UTC)
thread_title:  not understanding the end of the calculation 

message: The article on germs themselves defines them as equivalence classes of functions that agree on some neighborhood, but here they're defined essentially as the data for a specific power series. Shouldn't there be some note explaining that usage differs in this context (or whatever the case is)? 24.56.238.67 (talk) 20:44, 29 May 2024 (UTC)
thread_title:  germs as power series vs. as equivalence classes