Comments on: A surprising theorem in complex variables http://www.johndcook.com/blog/2009/05/05/a-surprising-theorem-in-complex-variables/ The blog of John D. Cook Sat, 11 Feb 2012 22:42:11 -0500 http://wordpress.org/?v=2.8.4 hourly 1 By: Tweets that mention A surprising theorem in complex variables — The Endeavour -- Topsy.com http://www.johndcook.com/blog/2009/05/05/a-surprising-theorem-in-complex-variables/comment-page-1/#comment-53331 Tweets that mention A surprising theorem in complex variables — The Endeavour -- Topsy.com Mon, 29 Nov 2010 17:09:51 +0000 http://www.johndcook.com/blog/?p=2191#comment-53331 [...] This post was mentioned on Twitter by beck, malena casas saucedo and Tom Cuchta, Analysis Fact. Analysis Fact said: A theorem of Nevanlinna http://bit.ly/5BpCWD // surprising theorem in complex analysis [...] [...] This post was mentioned on Twitter by beck, malena casas saucedo and Tom Cuchta, Analysis Fact. Analysis Fact said: A theorem of Nevanlinna http://bit.ly/5BpCWD // surprising theorem in complex analysis [...]

]]> By: John http://www.johndcook.com/blog/2009/05/05/a-surprising-theorem-in-complex-variables/comment-page-1/#comment-17000 John Tue, 05 May 2009 21:24:59 +0000 http://www.johndcook.com/blog/?p=2191#comment-17000 Mike: I ran across it reading <a href="http://www.amazon.com/gp/product/0444518312?ie=UTF8&tag=theende-20&linkCode=xm2&camp=1789&creativeASIN=0444518312" rel="nofollow">Nine Introductions in Complex Analysis</a>. I'm writing a review of the book. Mike: I ran across it reading Nine Introductions in Complex Analysis. I’m writing a review of the book.

]]> By: Michael Croucher http://www.johndcook.com/blog/2009/05/05/a-surprising-theorem-in-complex-variables/comment-page-1/#comment-16999 Michael Croucher Tue, 05 May 2009 21:22:17 +0000 http://www.johndcook.com/blog/?p=2191#comment-16999 Nice theorem! Did you come across it in the course of your work or simply stumble across it while browsing around? Nice theorem! Did you come across it in the course of your work or simply stumble across it while browsing around?

]]> By: Sue VanHattum http://www.johndcook.com/blog/2009/05/05/a-surprising-theorem-in-complex-variables/comment-page-1/#comment-16983 Sue VanHattum Tue, 05 May 2009 13:51:25 +0000 http://www.johndcook.com/blog/?p=2191#comment-16983 And my objection (I just tried to write it, and saw my mistake) was mixing up inputs and outputs also. I was thinking about equal on 5 domain values, but this is saying pick 5 output values which the functions achieve for the same set of input values. Interesting. And my objection (I just tried to write it, and saw my mistake) was mixing up inputs and outputs also. I was thinking about equal on 5 domain values, but this is saying pick 5 output values which the functions achieve for the same set of input values.

Interesting.

]]> By: John http://www.johndcook.com/blog/2009/05/05/a-surprising-theorem-in-complex-variables/comment-page-1/#comment-16977 John Tue, 05 May 2009 11:40:04 +0000 http://www.johndcook.com/blog/?p=2191#comment-16977 I hadn't state the theorem very clearly. Thanks for helping me see that. I updated the post. I hadn’t state the theorem very clearly. Thanks for helping me see that. I updated the post.

]]> By: Mark Reid http://www.johndcook.com/blog/2009/05/05/a-surprising-theorem-in-complex-variables/comment-page-1/#comment-16976 Mark Reid Tue, 05 May 2009 11:35:45 +0000 http://www.johndcook.com/blog/?p=2191#comment-16976 Good point. Thanks for clarifying. Good point. Thanks for clarifying.

]]> By: John http://www.johndcook.com/blog/2009/05/05/a-surprising-theorem-in-complex-variables/comment-page-1/#comment-16975 John Tue, 05 May 2009 11:28:18 +0000 http://www.johndcook.com/blog/?p=2191#comment-16975 Mark, the two functions could be constant without being equal. Say f(x) is constantly 5 and g(z) is constantly 6. Then f(z) = a and g(z) = a have the same solution set, namely the empty set, for a = 3 or any value of a other than 5 or 6. Mark, the two functions could be constant without being equal. Say f(x) is constantly 5 and g(z) is constantly 6. Then f(z) = a and g(z) = a have the same solution set, namely the empty set, for a = 3 or any value of a other than 5 or 6.

]]> By: Mark Reid http://www.johndcook.com/blog/2009/05/05/a-surprising-theorem-in-complex-variables/comment-page-1/#comment-16968 Mark Reid Tue, 05 May 2009 08:32:34 +0000 http://www.johndcook.com/blog/?p=2191#comment-16968 That is a strange theorem. I'm a bit puzzled by the bit where "f(z) and g(z) are constant or they are equal everywhere". This seems to suggest that when they are constant they are not equal everywhere. But surely they must be since they are both constant and equal on the points a_i? It is a surprising result though but, as you say, this sort of strangeness is often the case in complex analysis. The fact that the holomorphic functions are, thanks to the Cauchy-Riemann equations, are completely defined by their values on any small open set is also pretty weird. In some sense, this result of Nevanlinna's seems to be a generalisation of that well-known property in that you need only five points rather than an open set and that it holds for meromorphic functions. That is a strange theorem. I’m a bit puzzled by the bit where “f(z) and g(z) are constant or they are equal everywhere”. This seems to suggest that when they are constant they are not equal everywhere. But surely they must be since they are both constant and equal on the points a_i?

It is a surprising result though but, as you say, this sort of strangeness is often the case in complex analysis. The fact that the holomorphic functions are, thanks to the Cauchy-Riemann equations, are completely defined by their values on any small open set is also pretty weird.

In some sense, this result of Nevanlinna’s seems to be a generalisation of that well-known property in that you need only five points rather than an open set and that it holds for meromorphic functions.

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