Talk:Cauchy–Riemann equations

message: If f(z)=u(x,y)+ i v(x,y)satisfies cauchy reimann equation alongwith condition that u,v and all its 1st order patial derivatives are continuous, then the function is analytic ...........plz prove that --anon
: See Holomorphic functions are analytic. Oleg Alexandrov (talk) 21:30, 9 November 2005 (UTC)
thread_title: Holomorphic implies analytic

message: If a condition is "not necessarily sufficient" then it is not sufficient, surely?
I don't think the use of the more complex term is helpful. I'll take it out, unless someone else wants to, or wants to make a case for it.
David Young
I'm working on the survey article "complex analysis". I'm puzzled by the "necessary but not sufficient" language in this article. I'd like to rewrite this, because I don't think it's true. But I don't want to start another controversy inadvertently.
*If the Cauchy-Riemann equations are satisfied in a neighborhood, f(z) has a first derivative there. It doesn't matter whether the first partial derivatives are continuous functions or not ... we just have to be able to integrate them to get u and v.
*By the Cauchy-Goursat theorem, if f(z) has a first derivative in a neighborhood, it's analytic there. No requirement of continuity has to be imposed on the partials to do this proof, either.
*So the Cauchy-Riemann equations are in fact sufficient to prove analyticity, indirectly.
Am I missing something here?  DavidCBryant 15:48, 7 December 2006 (UTC)
replies: text: I've been thinking about this some more, and I think the current language (necessary but not sufficient) may have been written by somebody who only studied one textbook. Some authors develop the theory of functions by assuming that the partial derivatives in C-R are continuous, and eventually get around to Cauchy-Goursat. Other authors start off with complex integration, and prove Cauchy-Goursat first. Just a thought.  DavidCBryant 17:05, 7 December 2006 (UTC)
replies: text: I'm pretty sure the article is correct as it stands, as some sort of continuity assumption is required as well as the Cauchy-Riemann equations. For example: let f(x+iy) = 0 if one or both of x, y is zero, and f(x+iy) = 1 if neither of x, y is zero. Then, for this f, all the partial derivatives exist at the origin, and all are zero, so that the C-R equations hold. All this is for 'differentiability at a point'. of course, and you are probably right if we are considering C-R equations in a region. Madmath789 17:21, 7 December 2006 (UTC)
replies: text: Oops! Please forgive me for getting some facts mixed up. It's just been too long since I thought about some of this basic stuff. Proceeding directly from the C-R equations and trying to prove that holomorphic ==> analytic is the wrong way to go. Anyway, I probably shouldn't have even asked this question here in the first place, if I'd been thinking more carefully the other day.  DavidCBryant 20:04, 8 December 2006 (UTC)

text: The C-R conditions are necessary, but not sufficient. However, if a function is smooth enough (C^1 I think), the C-R conditions become sufficient. That is to say, the C-R conditions are necessary, and with a small extra assumption they become sufficient. Oleg Alexandrov (talk) 04:58, 15 May 2008 (UTC)

text: The statement is this: If f=u + iv, where u,v:R2->R, then provided u and v are differentiable at a point (x,y) (associated with z=x+iy) then f is analytic at z if and only if the Caucy-Riemann equations are satisfied.    This was kind of stated in the article, but not really so I'm going to modify the article to make this more clear. --67.193.128.233 (talk) 21:30, 13 April 2009 (UTC)
thread_title:  "Not necessarily sufficient" ??!! 

message: According to ru:Условия Коши — Римана, these equations were first published by D'Alambert in 1752. In 1777 Euler connected these equations to the analyticity of complex functions. Cauchy used these equations to construct the theory of functions in 1814. Riemann's dissertation on the theory of functions was published in 1851. Unfortunately, the Russian article does not have any references. Can anyone look into this?(Igny 15:09, 7 July 2006 (UTC)
replies: text: The relation has this interpretation:  \mathit{x} and \mathit{y} must be constant with respect to \bar z. This expresses the concept that an analytic function is "truly" a function of a single complex variable, rather than of a real vector.
It simply contradicts with x=.5(z+\bar z), y=.5(z-\bar z).
I find your use of a Russian language reference very unusual. Your references should be available to all; in this case those that speak English. Given that the per centage of English speakers that can also speak Russian is very small, using a Russian reference is not acceptable. Declan Davis (talk) 12:08, 17 September 2008 (UTC)
thread_title:  History 

message: There is a higher dimensional analogue of the CauchyRiemann equations involving a complex structure J^{\mu}_{\nu}
See e.g. http://en.wikipedia.org/wiki/Pseudoholomorphic_curve
It would be nice if this could be added here.
I think the polar form section would be better represented with the "r" terms. It suggests divide by zero problems in a way that could be confusing. The constants should simply be moved o the other side. You can see this form in Sarason's "Notes on Complex Function Theory." Comments? Are there any reasons that the current form is better? 
24.7.83.80 (talk) 09:25, 13 February 2008 (UTC)
thread_title:  More general form 

message: Is there a difference between analytic and holomorphic in this article? It sounds confusing to me.
: There is no difference, a holomorphic function of a complex variable is the same as an analytic function, although the latter term is more general, being used for functions of real variables too. Oleg Alexandrov (talk) 15:33, 8 August 2007 (UTC)
thread_title: Analyticity

message: I was trying to look for a derivation of the CR eqs. Is it possible to have it on the article? --Hamsterlopithecus (talk) 18:57, 7 October 2008 (UTC)
replies: text: There already is. Try the section on complex differentiability.  Δεκλαν Δαφισ   (talk)  21:25, 7 October 2008 (UTC)
replies: text: This is only necessity, not sufficiency.  --67.193.128.233 (talk) 21:32, 13 April 2009 (UTC)
thread_title:  Derivation 

message: Text says
:The hypotheses of Goursat's theorem can be weakened significantly.  If f = u+iv is continuous in an open set Ω and the partial derivatives of f with respect to x and y exist in Ω, then f is holomorphic (and thus analytic).  This result is the Looman–Menchoff theorem.
but does not recall Cauchy-Riemann equations. Isn't it misleading? --Bdmy (talk) 19:40, 1 January 2009 (UTC)
replies: text: This is more or less related but isn't any (distribution) solution of the homogeneous Cauchy-Riemann equations are infinitely differentiable, for {\partial \over \partial \overline{z}}u = 0 \Rightarrow \Delta u = 0 \Rightarrow u \in C^\infty (For example, by the elliptic regularity theorem)? -- Taku (talk) 02:22, 2 January 2009 (UTC)
replies: text: The last result quoted in the article follows from elliptic regularity.  Would it be worth mentioning this?  I'm not exactly sure how the other Goursat-like results compare with this one, since they deal only with classical notions of differentiability and continuity.  The corresponding classical regularity results obviously assume more differentiability than needed here. siℓℓy rabbit (talk) 03:34, 2 January 2009 (UTC)
thread_title:  Goursat section is not clear 

message: The assertion "irrotational + solenoidal" implies "to be a potential flow"  is not always true. It depends on the topology of the domain. For example 1/z does not have a potential because of the singularity in z=0. To correct the text it suffices to add that the domain U is simply connected. 80.174.196.107 (talk) 06:28, 6 May 2014 (UTC)
thread_title:  Physical interpretation 

message: The f-bar notation in the "physical interpretation" section is very poorly chosen, as it may be mistaken for complex conjugate.  Tkuvho (talk) 11:13, 12 April 2010 (UTC)
replies: text: It looks like the vector field needs to be the real and imaginary components of the complex conjugate of a holomorphic function, otherwise the Cauchy-Riemann equations do not give exactly the right conditions.  But the section does a rather poor job of explaining this. Sławomir Biały (talk) 12:33, 23 April 2010 (UTC)
thread_title:  f-bar 

message: The first sentence of the article:
:In mathematics, the Cauchy–Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations that provides a necessary and sufficient condition for a differentiable function to be holomorphic in an open set.
But if the function is differentiable, it is already holomorphic, by definition (and vice versa). I suppose the word "differentiable" should be omitted (or changed to smth like real and imaginary parts are differentiable as real-valued functions). Semifinalist (talk) 22:05, 30 August 2010 (UTC)
thread_title:  Differentiable as a prerequisite? 

message: Euler died in 1783.  Tkuvho (talk) 09:07, 8 December 2010 (UTC)
thread_title:  Euler 1797 ?? 

message: ...it looks like it is.  It's certainly continuous on the real line.  On which isn't it?  174.254.163.152 (talk) 02:20, 26 February 2011 (UTC)
thread_title: Not continuous at zero?

message: The article claims in more than one place that the differentiability of the corresponding R2 function and the C-R equations are necessary and sufficient to imply the complex function is holomorphic. However, is this claim not omitting that the partial derivatives must be continuous too in a neighbourhood of the point in question? This is not implied by differentiability in R2.
--94.169.117.101 (talk) 20:38, 16 April 2011 (UTC)
thread_title:  Missing condition? 

message: Should we by any chance add a section for the polar forms of C-R?
They are:
v_t = ru_r
and
u_t = -rv_r
and yield the following Laplace Equations:
r^2u_rr + ru_r + u_tt = r^2v_rr + rv_r - v_tt,
the proof for which is trivial.
I feel it would make the article feel more complete for people who are looking for it.71.234.166.64 (talk) 01:26, 26 August 2013 (UTC)
thread_title:  Proposal of a New Section 

message: :"Viewed as conjugate harmonic functions, the Cauchy–Riemann equations are a simple example of a Bäcklund transform."
This sentence does not make any sense.  You view the equations as functions, and then they become transforms.  ?????--345Kai (talk) 05:55, 24 September 2014 (UTC)
thread_title:  conjugate harmonic? 

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thread_title:  External links modified 

message: I think the generalization of the equations to higher dimensions is far more straightforward: the article on several complex variables makes no reference to the d-bar operator at all, and instead provides the much more elementary description using Wirtinger derivatives. I think we should use that here. The d-bar operator may provide the most general formulation, but is unnecessary for a simple extension to several variables.--Jasper Deng (talk) 10:04, 2 December 2017 (UTC)
thread_title:  No need to refer to the d-bar operator 

message: This article needs a more accessible lede. The first sentence is not understandable to the average reader, and it just goes deeper into the weeds from there. It needs a more general description for someone who did not major in math in college, and some reference to practical applications. I see that it is used to "check the differentiability of a complex function", to move a point function from cartesian to polar coordinates using D'Alembert–Euler condition and in triangular Mesh generation in computational fluid dynamics, for example to study flow around theoretical airfoil shapes without the time and expense of wind tunnel testing  . But this is probably better included by someone more knowledgeable than me. Dhaluza (talk) 17:36, 13 March 2021 (UTC)
thread_title:  Opaque Lede 

message: Where do they come from? They must be some kind of operator like object, given that a product of them produces the identity matrix. 23.93.95.115 (talk) 00:11, 9 January 2024 (UTC)
replies: text: The notation was bad.  The product isn't the identity matrix, but is a matrix whose square is minus the identity.  I have changed it to \sigma_1\sigma_2=J.  A representation of the Clifford algebra is \sigma_1=\begin{bmatrix}1&0\\0&-1\end{bmatrix} and \sigma_2=\begin{bmatrix}0&1\\1&0\end{bmatrix}.  Tito Omburo (talk) 11:49, 9 January 2024 (UTC)
thread_title:  Definition in Clifford  Algebra, what is $\sigma_1$ and $\sigma_2$? 

message: Hi, I'm not really a wikipedian and i have very little experience with editing wikipedia but I've noticed a mistake in the first image of this article (Cauchy-Riemann.svg). the two expressions on the bottom right shouldn't have a Z inside of the function since they describe the situation where you only multiply by z after applying the mapping. these expressions should be zdf(X) and df(X). I don't know how to fix this myself but I didn't see anyone else pointing the mistake. 109.253.198.52 (talk) 13:16, 22 April 2024 (UTC)
replies: text: Yes, this is an error. Tito Omburo (talk) 09:57, 24 April 2024 (UTC)
thread_title:  Mistake in the first image