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Find the flaw: Picard's Theorem


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#1 BMAD

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Posted 12 December 2013 - 03:04 AM

In complex analysis, an entire function is defined as a function which is infinitely differentiable at every point in C (for example: constants, polynomials, e^x, etc.). Picard's Theorem says that every nonconstant entire function f misses at most one point (i.e. f© = C or C-{x0}). For example, every nonconstant polynomial hits every point, and e^x misses only 0.
 
Now consider the function f(x) = e^(e^x). Since e^x is entire, f is also entire by the chain rule. But it misses 0 since the base e^y misses 0, and it misses 1 since the top e^x misses 0 so that e^(e^x) misses e^0 = 1. But by Picard's Theorem there can be only one missing point, so the two missing points must be the same. Therefore, 0 = 1.

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#2 Rainman

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Posted 12 December 2013 - 08:33 AM

Spoiler for


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#3 ThunderCloud

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Posted 15 December 2013 - 04:19 AM

Spoiler for


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#4 Rainman

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Posted 15 December 2013 - 09:24 AM

Spoiler for

It's a theorem in complex analysis, not in real analysis. Do you know about complex numbers?


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#5 BMAD

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Posted 15 December 2013 - 03:11 PM

Little Picard Theorem: If a function f : C → C is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point.

 

from wikipedia: http://en.wikipedia..../Picard_theorem


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#6 ThunderCloud

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Posted 15 December 2013 - 03:51 PM

 

Spoiler for

It's a theorem in complex analysis, not in real analysis. Do you know about complex numbers?

 

 

Ah, that was the part I missed. Thanks. :blush:


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