Talk:Cauchy-Riemann equations

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[edit] Holomorphic implies analytic

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)

[edit] "Not necessarily sufficient" ??!!

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)

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)
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)
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)

[edit] History

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))


I have edited the article, adding this information. Also I deleted the following nonsense

The relation has this interpretation: x and 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).

[edit] Why the Cauchy-Riemann Equations are Satisfied

An complex analytical function is continuous at a point Zo if and only if it's limit exist and is the same independent of what direction the limit takes. Here Cauchy is taking advantage of the fact that \lim_{\Delta x \rightarrow 0, \Delta y = o} is equivalent to \lim_{\Delta x = 0, \Delta y \rightarrow 0}.

—The preceding unsigned comment was added by 72.71.218.71 (talk • contribs) .

is that supposed to make any sense? --MarSch 14:32, 8 December 2006 (UTC)
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