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 Weak theorem about an almost identical function: PROOF Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008 11/14/2008, 01:11 PM bo198214 Wrote:Hi Dmitrii, though I didnt completely follow your proof it triggered coming up with a much more simplified proof. As a prerequisite we only need Picard's big theorem: Picard's big theorem applied to entire functions. Each entire non-polynomial function takes on every complex number with at most one exception infinitely often. First it can be shown that $J$ (as an entire function) does not even omit one value from $\mathbb{C}$. (I give the proof in a next post, though this assertion is not really necessary for the following conclusions.) We know that $J(k)=k$ for each $k\in\mathbb{Z}$, but by the above theorem for each $k\in\mathbb{Z}$ there have to be infinitely other $z\in \mathbb{C}\setminus \mathbb{Z}$ with $J(z)=k$. Hence $J(\mathbb{C}\setminus\mathbb{Z})=\mathbb{C}$. That means if we have a function $g(z)=f(J(z))$, where $f$ is a superexponential with singularities only at $\{z\in\mathbb{Z}:z\le -2\}$, then $g$ has singularities outside $\mathbb{C}\setminus\{z\in\mathbb{Z}:z\le -2\}$.Henryk, I like your proof. Indeed, it is shorter. Now, please, prove that some of singularities are in the right hand side of the complex plane. Small hint: From the asymptotic behavior, at 1

 Messages In This Thread Weak theorem about an almost identical function: PROOF - by Kouznetsov - 11/14/2008, 01:20 AM RE: Weak theorem about an almost identical function: PROOF - by bo198214 - 11/14/2008, 11:15 AM RE: Weak theorem about an almost identical function: PROOF - by bo198214 - 11/14/2008, 01:05 PM RE: Weak theorem about an almost identical function: PROOF - by Kouznetsov - 11/14/2008, 01:11 PM

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