Talk:Contour integration

message: In my university's adv. calculus course number 4 was called "Pacman rule". Is it popular imformal name or just local name?
: Hehe. Sounds like it's an informal (and local - I've never heard of it) name. Dysprosia 10:22, 3 September 2005 (UTC)
thread_title:  What's in a name? 

message: I'm going to go ahead and delete Example V which has a number of serious problems. Later on, if I have the time, I'll replace it with another example demonstrating the use of keyhole contours.
The problems I see with this example are the following.
1. There is a completely irrelevant remark at the beginning about the computer program MAPLE. This has nothing to do with with the choice of branch cut in this example. The reason for choosing this particular branch cut is that the avoids the integration contour.
2. It is claimed that the integral over the circular arcs tends to zero, but the only estimate actually shown in the article is the large R limit, and it's much less obvious that the integral over the small arc tends to zero as well.
3. In fact, its not even clear what small circular arc we're looking at since the picture of the contour does not match up with the computation a few lines later. Judging from the computation, the contour should contain two segments parallel to the negative x-axis, but instead the picture shows two lines emanating from the origin at an angle.
3. In the sixth line of displayed mathematics, the letter z is being used for three different things.
4. The word "or" is used indiscriminately in going from one line of computation to the next. In the first instance, it suggests equality, while in the second instance it seems to be saying "When we take the small-epsilon limit, we get..." Also, no justification is given for the limit taken in this line, which requires interchanging a limit with an improper integral.
5. I cant solve integral of (1+z)/z, where C is the right half of the circle |z| = 1 from z = -i to z = i.  — Preceding unsigned comment added by 174.117.247.151 (talk) 15:42, 7 December 2011 (UTC)
thread_title:  Embarrassing 

message: The first sentence says
:  "In complex analysis, contour integration is a method of evaluating certain integrals of real-valued functions along intervals on the real line that are not readily found by using only real variables."
This is an overly restrictive definition of what contour integration is, because it can be applied to complex-valued functions as well.  I would suggest rewording this to
:  "In complex analysis, contour integration is a method of evaluating certain integrals along closed paths in the complex plane.  This can be useful for evaluating integrals along the real line that are not readily found by using only real variables."
--Plasma g (talk) 16:49, 12 April 2008 (UTC)
thread_title:  The first sentence is inaccurate 

message: Some of the text in this page doesn't wrap properly around the figures in my browser (Firefox 1.0.4).  Anyone else see this problem/know how to fix it?
: Yeah I get the same problem.ACielecki 02:07, 12 April 2006 (UTC)
thread_title:  Bad wrapping of figures? 

message: I saw that there was no page for "Estimation lemma" which is linked to, so I created the page. It's the first one I've ever made so would appreciate help with it if you see room for improvement. For the example I used the same integral used in the example on this page, which I thought fit well.
Who creates the graphics? The "estimation lemma" could use an upper half-circle contour of radius 'a' and Γ near it to name the contour.
--Menorman 07:25, 23 June 2006 (UTC)
thread_title:  Added page linked to for "Estimation lemma", please check 

message: I added a redirect from Complex Integration to this page. Hope it's alright. Etnoy (talk) 22:31, 3 November 2008 (UTC)
thread_title:  Redirecting "Complex Integration" here 

message: A prod was stuck on saying 'Wikipedia is not meant to be a mathematical toolbox. That is true all right but I think all that is wrong with this article currently is the tone, That methods of contour integration is a notable subject in itself and most of the article could be kept. I think most of the problems could be removed by changing the tone so the descriptive bits don't say things like 'we will evaluate'. Dmcq (talk) 16:56, 23 July 2010 (UTC)
thread_title:  Not a mathematical toolbox 

message: Contour integral redirects here; yet the text appears to assume that the reader already knows what a contour integral is: the term is used without further explanation.  --Lambiam 21:06, 23 October 2011 (UTC)
replies: text: There did not seem to be a page that appropriately introduces the contour integral on Wikipedia, so I started such sections here. I believe that they are of admissible quality, but will require a bit of work to become what I would consider complete. In the future I could see these sections, as well as some of the general discussion of the intro, becoming their own page (such as Contour Integration). However, for now, considering their length I think this location is appropriate, especially considering that all related links currently direct readers to this page. Particular improvements I hope to do to these sections are to lengthen the conceptual discussion and add more references. Also, I would like to see the subsection Methods of contour integration#Generalization of the Riemann Integral to be mathematically precise (and thus longer). Further results such as independence of path should be included, as well as a short discussion of other results which should link to appropriate pages (such as Cauchy integral formula and Cauchy integral theorem).Brent Perreault (talk) 18:29, 17 December 2012 (UTC)
thread_title:  What is a contour integral? 

message: At the bottom of Contour integrals, it states: That is, the result is independent of the curve chosen.[6] In the case where the real integral on the right side does not exist the integral along γ is said not to exist. 
I assume it was meant to be independent of PARAMETRIZATION, since the example just a little bit down shows the reader that the integral of 1/z over the complex unit circle is 2πi, contradicting the statement that there is an independence of PATH.
Svein Olav Nyberg (talk) 06:02, 20 March 2013 (GMT+1)
Agreed! The result does not depend on restriction to the unit circle.  See comment below.
David in Cincinnati (talk) 22:32, 11 January 2020 (UTC)
thread_title:  Independent of path? 

message: I think this would be useful, now some of the steps are a bit confusing.  I've seen this convention in a couple books and also in some wikipedia articles.  — Preceding unsigned comment added by 94.112.136.34 (talk) 20:43, 7 June 2014 (UTC)
thread_title:  Use log for complex logarithm and ln for the real one 

message: The paragraph “The parametrization of a curve ….”  is a little confusing.  I think that more succinct would be:
A directed smooth curve can be defined as a smooth curve together with direction defined by designation of the image of a or b in a specific parametrization as initial point.  Then the parametrization defines ordering of points along the curve.  For example, if the image of b is the initial point, then x is further along the curve than y if x < y.  With the initial point fixed, the ordering is independent of the parametrization chosen.
David in Cincinnati (talk) 22:02, 11 January 2020 (UTC)
replies: text: This is a really strange way of introducing the notion.  You don't need to insist on injectivity or equality of the derivative at the endpoints.  The whole section could use some streamlining.  Textbooks I've seen usually just rely on something like piecewise smoothness of the curve.  And I have no idea why there's even a digression into the bit about ordering the points along the curve.  –Deacon Vorbis (carbon • videos) 22:43, 11 January 2020 (UTC)
thread_title:  Directed smooth curves 

message: There’s no reason to restrict the the contour integral of  to the unit circle.  Simply let , etc.  That the result (2𝝅i) is independent of the radius of the circle is vital to other applications and also is a simple illustration of how such an integral is largely independent of path taken by the closed contour.  In fact, it would be just as simple to evaluate the more general:
     \oint_C z^j\,dz ,   
which vanishes for all integer j except j = -1.  This could then be a good point to introduce the idea of “n-th order poles” and to the central feature of contour integration, namely the ease in deforming closed contours with minor, if any, change in the integral result.  The article would then be more self-contained without the current heavy reliance on links to other articles, such as the "residue theorem".  A good follow-up example could then be evaluation of the Dirichlet integral, 
     \int_{-\infty}^\infty \frac{sin(kx)}{x}\,dx,
one of the simplest, yet most important application of contour integration. 
  
David in Cincinnati (talk) 22:23, 11 January 2020 (UTC)
thread_title:  Example 

message: I noticed that the page calls "smooth curve" what is usually called "_simple_ (smooth) curve" or path, or Jordan curve, that is the image of an injective (smooth) map from an interval or a circle -or a parametrization thereof. This is unusual and will surely confuse some readers. Curves appear in all areas of mathematics, often without the injectivity assumption, so the sensible terminology choice is to add the adjective "simple" every time we require it, and this is what texts on complex analysis do -at least all those i have read. Note also that the injectivity assumption is not necessary to define the line integral, thus attentive readers will also be confused by that superfluous assumption.
Also, the wording in the paragraph defining "smooth curve" is awkward.
In conclusion i think the editors of this page should remove the assumption of injectivity, just call "smooth curve" what everyone calls that, and if possible improve the phrasing of the paragraph defining "smooth curve". Plm203 (talk) 10:14, 23 June 2023 (UTC)
thread_title:  Why the unusual name for a simple curve ? And why the restriction to simple curves ? 

message: The circle-integral-sign comes out of the blue. It needs to be explicitly defined. Kontribuanto (talk) 11:50, 12 April 2024 (UTC)
thread_title:  definition of circle-integral-sign needed