Similar to the kouznetsov method I also consider a contour integral. But instead of taking THE shape of the contour and searching for the values on them , i give the values and search THE shape.
Im close to proving this I think.
This also implies that Im more optimistic about the traditional kouznetsov-cauchy method.
I call it KTC-method which stands for Kouznetsov-Tommy-Cauchy method.
Basicly the contour is split in 4 parts where the upper and lower are straight horizontal lines just like in Kouznetsov's method.
The given values and the searched shape is thus in the almost vertical contourparts.
Just like in Kouznetsov's method the precision is increased by " stretching " the contour to imaginary infinity.
The values satisfy f(z*) = f*(z) , where * stands for complex conjugate , this to assure that we find a locally real-analytic solution.
The main reason to support this and the Original Kouznetsov method is that the contour and its values only need to be continuous to imply that the interior is analytic.
Then secondly by the functional equation and analytic continuation we get an analytic strip in both the upper and lower halfplane where we have an analytic sexp.
These strips start at Im > 0 resp Im < 0 and are at least as high/low as Im(uppercurve ( the straith hor. line of the contour ) ) resp Im(lowercurve).
Values should be chosen such that the path of the contour follow asymptotically f(a +/- bi) where b >> a.
( SO that we get close the fixpoint and get higher precision )
Preliminary results are optimistic.
Pseudoperiod appears to become visible.
Im not sure how do an update procedure to improve initial guesses and the alike , but even without that it seems to work.
Excited
regards
tommy1729
Im close to proving this I think.
This also implies that Im more optimistic about the traditional kouznetsov-cauchy method.
I call it KTC-method which stands for Kouznetsov-Tommy-Cauchy method.
Basicly the contour is split in 4 parts where the upper and lower are straight horizontal lines just like in Kouznetsov's method.
The given values and the searched shape is thus in the almost vertical contourparts.
Just like in Kouznetsov's method the precision is increased by " stretching " the contour to imaginary infinity.
The values satisfy f(z*) = f*(z) , where * stands for complex conjugate , this to assure that we find a locally real-analytic solution.
The main reason to support this and the Original Kouznetsov method is that the contour and its values only need to be continuous to imply that the interior is analytic.
Then secondly by the functional equation and analytic continuation we get an analytic strip in both the upper and lower halfplane where we have an analytic sexp.
These strips start at Im > 0 resp Im < 0 and are at least as high/low as Im(uppercurve ( the straith hor. line of the contour ) ) resp Im(lowercurve).
Values should be chosen such that the path of the contour follow asymptotically f(a +/- bi) where b >> a.
( SO that we get close the fixpoint and get higher precision )
Preliminary results are optimistic.
Pseudoperiod appears to become visible.
Im not sure how do an update procedure to improve initial guesses and the alike , but even without that it seems to work.
Excited
regards
tommy1729