Talk:Morera's theorem

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The proof provided here is unclear, the function F(z) should be defined by picking any point z_0 in D and letting F(z) be the path integral of f(z) along any path from z_0 to z. The arguments presented in the given proof show why this is well defined. I would fix it myself but. . . i'm not too hot with the math editing. —Preceding unsigned comment added by 66.75.254.51 (talk • contribs)

There should be a link to the article about Morera somewhere in this article (or at least his name if he isn't notable enough for an article). -- Obradović Goran (talk 19:22, 21 June 2007 (UTC)

I've now added a biographical stub article about Morera Richard holt 10:08, 14 September 2007 (UTC)Richard Holt


the proof given in article is actually slightly longer than necessary and requires the domain D be connected. what is needed is just that f has an anti-derivative locally in D, or even in a open disk in each connected component of D (where there's no well-definedness issue). the claim then follows from the identity theorem. Mct mht 21:27, 6 August 2007 (UTC)

IIRC, Morera's theorem actually characterizes holomorphy. the assumption that the integral of f along every closed path (therefore having an anti-derivative on all of its domain) is too strong for this (e.g. 1/z in some annulus region centered at 0). if that sounds about right to folks here, i might modify the article accordingly. Mct mht 18:45, 8 August 2007 (UTC)

Does this theorem have the right name? The Morera theorem that I learned in class is actually the one found on the mathworld page: http://mathworld.wolfram.com/MorerasTheorem.html While they may be basically equivalent, it is confusing to state a theorem in a non-standard way that is not obviously the same. Newlyformed 06:54, 14 August 2007 (UTC)

I would prefer to go back to the older statement, with the integral. Charles Matthews 07:01, 14 August 2007 (UTC)
they ain't the same though. the old version says that if f has an anti-derivative everywhere on its domain (or, equivalently, has zero integral along every closed curve in the domain), then it is holomorphic. but that is trivial and, far as i know, not Morera's theorem. Mct mht 15:04, 14 August 2007 (UTC)
well, it's getting reverted to the previous version. one might wanna revert further back though, as right now the article is not quite coherent. there's really nothing complex-analytic about this formulation of the theorem, the argument would be a verbatim copy from the real variable case. Mct mht 22:28, 13 September 2007 (UTC)
The version of Morera's theorem now in the article is the same version stated in Rudin's book, in Ahlfors' book, and in Conway's book. There's nothing trivial about it: "holomorphic" means "complex differentiable", and it's a relatively deep result that a complex function that locally possesses an anti-derivative should have a derivative too. The corresponding statement in the real case would be "any vector field on R2 whose line integral around every closed loop is zero must be differentiable", and this is completely false. The proof in the article makes the theorem seem somewhat easy, but only because it assumes that holomorphic functions are analytic. Jim 19:51, 15 September 2007 (UTC)

"A verbatim copy from the real-variable case"?? There's no real-variable counterpart of this result. This works for complex variables; NOT for real variables. Michael Hardy 20:10, 15 September 2007 (UTC)

"the integral of a continuous function is differentiable", that part of the claim is a word by word copy of a similar fact for functions on the real line and is entirely trivial. of course one would then invoke differentiability implies analyticity in the complex-variable case. the point ought to be made that this formulation of the theorem does not use the full strength of the fact that holomorphy means analyticity. having zero integral along a every closed path, i.e. possessing an anti-derivative in the complex variable sense everywhere on its domain, is a far more restrictive condition than being holomorphic. Mct mht 23:56, 15 September 2007 (UTC)
You seem to be confused about something, though maybe I'm misinterpreting. The theorem states that a function which can be integrated is differentiable. It is not part of the hypothesis of the theorem that f is the integral of a continuous function. Jim 08:30, 21 September 2007 (UTC)
yes, you are misinterpreting. Mct mht 19:43, 21 September 2007 (UTC)

[edit] Converse to Cauchy's Theorem

Mct, I strongly disagree with you removing the text about Morera's theorem being a converse to Cauchy's theorem. Every complex analysis book that I have mentions this immediately before stating the theorem. The fact is that the two theorems are converses when the domain is simply connected, and this is by far the most common case for applications. Jim 08:23, 21 September 2007 (UTC)

they are mutually converses when the domain is simply connected means it's ok to imply they are mutually converses in general, which is clearly untrue? news to me. Mct mht 19:43, 21 September 2007 (UTC)

From page 209 of Rudin "Real & Complex Analysis":

The Cauchy theorem has a useful converse:
[statement of Morera's theorem]

From page 86 of Conway's "Functions of One Complex Variable":

Cauchy's Theorem and Integral Formula is the basic result of complex analysis. With a result that is so fundamental to an entire theory it is usual in mathematics to seek the outer limits of the theorem's validity. Are there other functions that satisfy \textstyle\int_\gamma f = 0\,\! for all closed curves γ? The answer is no as the following converse to Cauchy's Theorem shows.
[statement of Morera's theorem]

The thing is, the simply connected case really is the heart of the theorem. Even when the domain is the unit disc, it is an extremely unexpected result that an integrable function is also differentiable. Jim 21:54, 21 September 2007 (UTC)

put it back if you want. i modified the statement to "the converse of the theorem is not true in general, although [paraphrasing] it is, for instance, if the domain is simply connected (Cauchy's theorem)." and you wanna insist that one should say Cauchy's theorem is "a" converse? fine.
Re "The thing is, the simply connected case really is the heart of the theorem." says you? what theorem? Cauchy's theorem? the heart of theorems of Cauchy or Goursat type is that one wanna find conditions under which the integral of a holomorphic functions over a closed curve does vanish. the domain being simply connected or convex would be among these conditions. they in turn lead to Cauchy's theorem and the Cauchy integral formula. another condition would be consider just triangles, as the last part of the article states. this is Goursat's lemma and gives a characterization of holomorphy.
Re "it's an extremely unexpected result that an integrable function is also differentiable." "extremely unexpected"? the class of integrable functions is a proper subclass of holomorphic functions. if you find whatever it is you find "extremely unexpected" extremely unexpected, holomorphic functions being analytic in general must surprise the friggin hell outta you. fine, i am impressed. Mct mht 03:11, 22 September 2007 (UTC)
Yes, that's what I meant by unexpected. Morera's theorem is unexpected because it's unexpected that holomorphic functions are analytic. Jim 07:03, 22 September 2007 (UTC)