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 Kneser's Super Logarithm bo198214 Administrator Posts: 1,391 Threads: 90 Joined: Aug 2007 11/23/2008, 01:00 PM (This post was last modified: 11/23/2008, 01:03 PM by bo198214.) As the last step we use the Riemann mapping theorem, to map the area $L$ biholomorphically to the upper halfplane, which maps the boundary to the real axis. By some reason (to be explained) the corresponding function $\Psi$ indeed satisfies the Abel equation. But we do that in two steps first we map $L$ to the unit disk $\mathbb{E}$ via $\alpha: L\leftrightarrow\mathbb{E}$ and then we map $\mathbb{E}$ to the upper (open) halfplane $\mathbb{H}$ via $\beta:\mathbb{E}\leftrightarrow\mathbb{H}$. The existence of $\alpha$ is guarantied by the Riemann mapping theorem. The translation $z\mapsto z+c$ maps $L$ to $L$ without having a fixed point, hence the corresponding mapping in $\mathbb{E}$, $\tau:\alpha(z)\mapsto \alpha(z+c)$ maps the unit disk into the unit disk without a fixed point. It is well known that such a function needs to be linear fractional, i.e. of the form $\tau(z)=\frac{d_1 z+d_2}{d_3 z +d_4}$. Kneser shows that must be parabolic, that means it has only one fixed point $p$ at the boundary of $\mathbb{E}$. To map the unit disk to the upper half plane we can again use a linear fractional transformation $\beta(z)$. There are enough parameters to chose it such that $p$ is mapped to infinity and such that $\beta(\tau(z))=z+1$. Now we can define $\Psi(z)=\beta(\alpha(\psi(z))$ which has the property $\Psi(e^z)=\beta(\alpha(\psi(z)+c))=\beta(\tau(\alpha(\psi(z)))=1+\Psi(z)$. and is biholomorphic on the interior of $H_{-1}\cup H_0\cup H_1$. The interior of $H_{-1}\cup H_0\cup H_1$ is mapped to some area bordering on the real line. The boundary on the real axis is hence mapped to the real axis. More precisely the interval $(\log(\Re( c)),e^{|c|})$ is mapped to the real axis. By the Schwarz reflection principle it can be continued to the complex conjugate of $H_{-1}\cup H_0 \cup H_1$ especially it is analytic on $(\log(\Re( c)),e^{|c|})$ and can from there be continued to a vicinity of the whole real axis by $\Psi(e^x)=\Psi(x)+1$ and $\Psi(\ln(x))=\Psi(x)-1$. « Next Oldest | Next Newest »

 Messages In This Thread Kneser's Super Logarithm - by bo198214 - 11/19/2008, 02:20 PM RE: Kneser's Super Logarithm - by bo198214 - 11/19/2008, 03:25 PM RE: Kneser's Super Logarithm - by sheldonison - 01/23/2010, 01:01 PM RE: Kneser's Super Logarithm - by mike3 - 01/25/2010, 06:35 AM RE: Kneser's Super Logarithm - by sheldonison - 01/25/2010, 07:42 AM RE: Kneser's Super Logarithm - by mike3 - 01/26/2010, 06:24 AM RE: Kneser's Super Logarithm - by sheldonison - 01/26/2010, 01:22 PM RE: Kneser's Super Logarithm - by mike3 - 01/27/2010, 06:28 PM RE: Kneser's Super Logarithm - by sheldonison - 01/27/2010, 08:30 PM RE: Kneser's Super Logarithm - by mike3 - 01/28/2010, 08:52 PM RE: Kneser's Super Logarithm - by sheldonison - 01/28/2010, 10:08 PM RE: Kneser's Super Logarithm - by mike3 - 01/29/2010, 06:43 AM RE: Kneser's Super Logarithm - by bo198214 - 01/26/2010, 11:19 PM RE: Kneser's Super Logarithm - by sheldonison - 01/27/2010, 07:51 PM RE: Kneser's Super Logarithm - by bo198214 - 11/22/2008, 06:11 PM RE: Kneser's Super Logarithm - by bo198214 - 11/23/2008, 01:00 PM

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