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.

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.