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open problems survey
#4
Conjecture
Let be the Carleman matrix of (truncated to N rows and columns), , real.
Then the set of eigenvalues of converges to the set for in the sense that there exist an enumeration of the Eigenvalues of such that for each .

Discussion
This is about the function shifted by .
The fixed point 0 is a singularity for (for non-natural ), so has to be developed at the different point .
In the particular case we have the fixed point at 0 and the first derivative is . So the Carleman matrix is triangular and we can solve it exactly, getting .
The conjecture is again about the independence of the matrix function method with respect to the development point.

can even be developed at the fixed point 0 in the particular case . However in this case except and regular iteration can not be applied, which makes sense as can for most t not be developed at 0.
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Messages In This Thread
open problems survey - by bo198214 - 05/17/2008, 10:03 AM
Exponential Factorial, TPID 2 - by andydude - 05/26/2008, 03:24 PM
eigenvalues of Carleman matrix for (x+s)^p-s, TPID 3 - by bo198214 - 06/29/2008, 12:36 PM
Existence of bounded b^z TPID 4 - by bo198214 - 10/08/2008, 04:22 PM
A conjecture on bounds. TPID 7 - by andydude - 10/23/2009, 05:27 AM
Logarithm reciprocal TPID 9 - by bo198214 - 07/20/2010, 05:50 AM
RE: open problems survey - by nuninho1980 - 10/31/2010, 09:50 PM
Tommy's conjecture TPID 16 - by tommy1729 - 06/07/2014, 10:44 PM
The third super-root TPID 18 - by andydude - 12/25/2015, 06:16 AM

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