Talk:Contour integration

What's in a name?

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)[reply]

Embarrassing

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)[reply]

In summary, I find this example to be an embarrassing example of poor mathematical writing. I haven't read the rest of the article, but if it contains the same basic errors, then I think it may need attention from an expert. 208.46.240.2 (talk) 20:42, 8 August 2011 (UTC)[reply]

I have restored it, minus the Maple sentence (which is ridiculous). While I agree with most of your points, I do not agree that this makes it unhelpful; indeed, I found the section quite helpful, and think that cleanup is more appropriate than deletion. Leaving it here with a discussion here directing somebody to clean it up, change the image, etc., will be more helpful than removing it, since the core content is quite good. --74.196.121.135 (talk) 19:51, 11 August 2011 (UTC)[reply]

Yes. Please don't delete things based on what they OMIT. That makes no sense whatsoever. Add to them. Harsimaja (talk) 01:29, 24 December 2011 (UTC)[reply]

The first sentence is inaccurate

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)[reply]

there's a minor error

When it gives cos[theta]=(1/2)(z+1/z), it first states cos[theta] as (1/2)e^{i theta) +e^(-i theta). The 1/2 should multiply both exponents. Minor error, but worth fixing whenever anyone reads this that knows how to edit the math. And when you fix it, please delete my comment =)

Nelson

I thought I'd mention that this appears to have been fixed by now.

Vote for new external link

Here is my site with contour integration example problems. Someone please put this link in the external links section if you think it's helpful and relevant. Tbsmith

http://www.exampleproblems.com/wiki/index.php/Complex_Variables#Complex_Integrals

Bad wrapping of figures?

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)[reply]

Added page linked to for "Estimation lemma", please check

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)[reply]

Redirecting "Complex Integration" here

I added a redirect from Complex Integration to this page. Hope it's alright. Etnoy (talk) 22:31, 3 November 2008 (UTC)[reply]

NPOV/Accuracy

See [1]. The tags should remain until this is clarified. Headbomb {^ταλκ_{κοντριβς} – WP Physics} 01:28, 22 February 2010 (UTC)[reply]

Nonsense! Count Iblis (talk) 01:41, 22 February 2010 (UTC)[reply]

The sources seem to be on-topic, professionally published sources. As described at WP:SCG, this article seems to footnote several general sources on the first line that can be used to verify statements lower down.

Unless there are specific concerns, I don't see any glaring difficulties with the sources. — Carl (CBM · talk) 02:48, 22 February 2010 (UTC)[reply]

Alright, CBM's vouching is good enough for me. Headbomb {^ταλκ_{κοντριβς} – WP Physics} 03:10, 22 February 2010 (UTC)[reply]

Actually, CMB vouching or not should be completely irrelevant. This article contains a large number of derivations which themselves cannot be directly sourced. Of course, it would be plaigiarism if they could be directly sourced. I think this article is fine. But according to Headbomb's logic used in the infraparticle page discussion, this article should be very problematic. People who are not fluent in maths cannot verify this article from the sources.

B.t.w., I do think one needs to explain the residue at infinity either by giving a source for that specific statement or an argument that explains why it is defined in that way (no source would be ok with me in the latter case). I do myself understand this, but I know that many people who know the basics of complex analysis do not. Count Iblis (talk) 13:11, 22 February 2010 (UTC)[reply]

Please note that the article's verifiability is perfectly adequate. Charles Matthews (talk) 13:18, 22 February 2010 (UTC)[reply]

I agree 100%. Count Iblis (talk) 13:45, 22 February 2010 (UTC)[reply]

Not a mathematical toolbox

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)[reply]

Nothing wrong with adding the occasional 'toolbox' article - the book-form Britannica includes such articles too - it helps more people than are grievously offended by its existence, so this person can lighten up. Same goes for the article on vector calculus in various coordinate systems. I'm thinking of adding a similar one myself. Harsimaja (talk) 01:27, 24 December 2011 (UTC)[reply]

What is a contour integral?

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)[reply]

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)[reply]

I agree. I think the article could be expanded further, resulting in a more complete description of contour integration accessible to the uninitiated. For example, a very simple proof of Cauchy integral/residue theorem could be included, obviating link to a more complete, yet abstruse article. The result could be a good introduction to the beauty of the basics and application of this field.

David in Cincinnati (talk) 22:46, 11 January 2020 (UTC)[reply]

Independent of path?

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)[reply]

Use log for complex logarithm and ln for the real one

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)[reply]

Requested move 4 July 2017

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: page moved. (closed by non-admin page mover) —Guan aco 07:33, 12 July 2017 (UTC)[reply]

Methods of contour integration → Contour integration – Per WP:PRECISION, this article is primarily about contour integration, and this is the main article about it. Specific methods are discussed, but they're not the main point of the article. Deacon Vorbis (talk) 18:42, 4 July 2017 (UTC)[reply]

The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

Directed smooth curves

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)[reply]

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)[reply]

Example

There’s no reason to restrict the the contour integral of $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/z⁠$ to the unit circle. Simply let $z (t) = re it$ , 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)[reply]

Why the unusual name for a simple curve ? And why the restriction to simple curves ?

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)[reply]

definition of circle-integral-sign needed

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

Fundamental definition is false

The section As a generalization of the Riemann integral contains this sentence:

"The integral over the curve is the limit of finite sums of function values, taken at the points on the partition, in the limit that the maximum distance between any two successive points on the partition (in the two-dimensional complex plane), also known as the mesh, goes to zero."

This is blatantly false as stated.

I hope someone familiar with integration of complex functions can correct this erroneous statement.