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 Kneser's Super Logarithm bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 11/19/2008, 03:25 PM (This post was last modified: 11/19/2008, 03:30 PM by bo198214.) As mentioned in the title of his paper Kneser originally seeks for an analytic solution $\phi$ of the functional equation $\phi(\phi(x))=e^x$ (This functional equation was considerably discussed at the conference of the German mathematician's association, October 1941 in Jena. So there the idea was born to construct an analytic solution.) As is very well-known the construction of a half-iterate can be reduced to the solution of the Abel equation, which Kneser gives here in a bit more generality: $\psi(f(x))=\psi(x)+\beta$. If one has such a solution $\psi$ then it is easy to construct the half iterate by: $\phi(x)=\psi^{-1}(\frac{\beta}{2}+\psi(x))$ because then \begin{align*} \phi(\phi(x)) &=\psi^{-1}(\frac{\beta}{2}+\psi(\psi^{-1}(\frac{\beta}{2}+\psi(x)))\\ &=\psi^{-1}(\frac{\beta}{2}+\frac{\beta}{2} +\psi(x))\\ &=\psi^{-1}(\beta+\psi(x))\\ &= \psi^{-1}(\psi(f(x))\\ &=f(x) \end{align*} Closely related to the Abel equation is the Schroeder equation: $\chi(f(x))=\gamma \chi(x)$ If one has a solution of the Schroeder equation, then $\psi(x)=\log(\chi(x))$ is a solution of the Abel equation with $\beta=\log(\gamma)$, because: $\psi(f(x))=\log(\chi(f(x))=\log(\gamma)+\log(\chi(x))=\log(\gamma)+\psi(x)$ And one can also derive the fractional iterates directly from the Schroeder function by: $f^{t}(x)=\chi^{-1}(\gamma^t\chi(x))$. The good thing is now that for holomorphic functions with fixed point $c$ and $a:=f'( c)\neq 0,1$ there is always the so called principal Schroeder function (holomorphic in a vicinity of $c$) which satisfies the Schroeder equation for $\gamma=a$ and which satisfies $\chi'( c)=1$. There are two possibilities to construct this principal Schroeder function. First by development of the powerseries in $c$ and chosing the coefficients of $\chi$ such that they satisfy the Schroeder equation. And second by the limit $\chi(x)=\lim_{n\to\infty} \frac{f^n(x)-c}{a^n}$ As you can easily verify if $\chi$ is a solution of the Schroeder equation then $b\chi(z)$ is also a solution for any constant $b$. They are called "regular" (by Szekeres) if $\chi$ is the principal Schroeder function. Those regular solutions are characterized by that $\chi^{-1}(\gamma^t \chi(x))$ is holomorphic in $c$. As a first step Kneser computes this principal Schroeder function $\chi$ of $\exp(z)$ at $\exp$'s first fixed point $c\approx0.318+1.337i$ in the upper half plane, the fixed point nearest to the real axis. $\chi(z)$ is however not real on the real axis, so he determines some mapping properties of $\chi$ and later manipulates $\chi$ to become real and analytic on the real axis. « 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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