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Topic: PICARD'S THEOREM PROOF THAT 0 = 1 (Read 1025 times) |
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jw666
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I stumbled upon your website while doing research on the picard theorems in complex analysis. I believe your proof that 0=1 does not work because Picard's Theorem is for complex numbers and the exponential function in the complex plain is periodic and does infact assume ALL values but 0 (including the negative real axis). I have attached some work I did after reading your riddle to show that e^(e^x) can equal 1 when x is a complex number.
-Daniel
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Icarus
wu::riddles Moderator
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Boldly going where even angels fear to tread.
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Re: PICARD'S THEOREM PROOF THAT 0 = 1
« Reply #1 on: Nov 25th, 2006, 7:43am » |
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That is the answer. There are other values of w than w = 0 for which ew = 1. Since the inner exponential takes on these other values w, the outer exponential takes on the value 1.
The puzzle depends on the fact that people are used to thinking of ez as being 1-to-1, since the restriction of it to the real numbers is. If ez were 1-to-1, then there indeed would be no way for the outer exponent to ever = 1.
It really is a clever "proof", and unfortunate that most forum visitors have never heard of Picard's theorem, and cannot appreciate it.
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| « Last Edit: Nov 25th, 2006, 7:44am by Icarus » |
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"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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