Well, @MphLee - thank you as well for your kind answer!
Well, I just put a vague idea... If we separate the powerseries of the exp() in two partial series, then the partial series are as well interesting functions (sinh() and cosh() ) and if in the two partial series we change each second coefficient's sign we have sin() and cos() instead, which relate to sinh() and cosh() by using imaginary instead real arguments. As said: just a vague idea, perhaps there might something similar interesting come out ... if researched effectively...
But the central topic is: 1) do we know a functional bound for the coefficients depending on the index \( k\) at all? (I.N. Baker said in his 1958 article: "we have no apriori knowledge about their growthrate" and I've never seen some estimate of this) 2) can we use a known summation method, for instance Borel-summation, such that we have also a formally proved arbitrarily-approximatable procedure, or that we can build a new one based on our knowledge of that specific growth-rate. - - - (Btw.: the separation into four partial series is not really important here: the limiting is of course valid for the unsegmented series as well) A big plan, likely too big for me, but my heuristic is simply too nice...
Nothing to problematize. I thought only about something like for a complete beginner, how such summations work - in principle, with examples. If sometimes someone liked that idea he/she might come back to this :-)
Sorry, this was meant chap 3 (in my article).
And while I'm scanning my old hobby-treatizes I think there are some even better workouts, however I never completed a stand-alone text on this subject.
This subject can really make one dizzy. I even thought to find a summation for \( su= 10 - 10^{10} + 10^{10^{10}} - \,^4 10 + \,^5 10 \pm \cdots \pm \) to a finite value, maybe a completely dizzy procedure having googol and googolplex like numbers already in the very first terms of the series... :-)
Ehhmm - btw. the Knopp-book is also existent in english translation, and maybemaybemaybe even accessible online, don't know. (The german version of his book was/is accessible via digitizing-center of university of Göttingen, maybe this is perhaps a hint...)
- - - - -
To make the matter now a bit less prominent: having strong loss of energy since several monthes now I'm only skimming through my early elaborations and word-docs and if I find something worth to be looked at, which I did not present earlier here in the forum (but which is related to our matter), I think I'll add it "to the pipeline" for someone who might be interested by chance (or has a knack for historical matters)... Hmm, maybe a subforum for such thoughts might even be more appropriate...
Kind regards -
Gottfried
(07/21/2022, 12:15 PM)MphLee Wrote:(07/19/2022, 10:19 PM)Gottfried Wrote: However, having 4 partial sequences, this might moreover reflect some effect in the question of negative \(z \) and imaginary \( z \). This might then be interesting in its own...Can we derive from this regular behavior modulo 4 that modulo 4 and the coefficient themselves, even if they produce divergence, are following some kind of law, have meaning?
It is that that you are pointing to?
Well, I just put a vague idea... If we separate the powerseries of the exp() in two partial series, then the partial series are as well interesting functions (sinh() and cosh() ) and if in the two partial series we change each second coefficient's sign we have sin() and cos() instead, which relate to sinh() and cosh() by using imaginary instead real arguments. As said: just a vague idea, perhaps there might something similar interesting come out ... if researched effectively...
But the central topic is: 1) do we know a functional bound for the coefficients depending on the index \( k\) at all? (I.N. Baker said in his 1958 article: "we have no apriori knowledge about their growthrate" and I've never seen some estimate of this) 2) can we use a known summation method, for instance Borel-summation, such that we have also a formally proved arbitrarily-approximatable procedure, or that we can build a new one based on our knowledge of that specific growth-rate. - - - (Btw.: the separation into four partial series is not really important here: the limiting is of course valid for the unsegmented series as well) A big plan, likely too big for me, but my heuristic is simply too nice...
(07/21/2022, 12:15 PM)MphLee Wrote: Anyways... it is a very rich zoo of summation methods...:-) Yes, sure. Has been too much for me to investigate seriously when I had also to do my job. And to accompany my family. And to be pappa for my child in home. :-)
Quote:Gottfried Wrote:(...) and if interested I might prepare some zoom- or skype-meeting to show that at work using Pari/GP and my matrices... :-)I'd be happy but I'm not sure I have the prerequisites and also the time needed to apply on this (...)
Nothing to problematize. I thought only about something like for a complete beginner, how such summations work - in principle, with examples. If sometimes someone liked that idea he/she might come back to this :-)
Quote:Btw... chapter 5 is the "Online res(S)ources" part... and I suspect you are pointing me to the geman languange bibliography items...are you?
Sorry, this was meant chap 3 (in my article).
And while I'm scanning my old hobby-treatizes I think there are some even better workouts, however I never completed a stand-alone text on this subject.
This subject can really make one dizzy. I even thought to find a summation for \( su= 10 - 10^{10} + 10^{10^{10}} - \,^4 10 + \,^5 10 \pm \cdots \pm \) to a finite value, maybe a completely dizzy procedure having googol and googolplex like numbers already in the very first terms of the series... :-)
Ehhmm - btw. the Knopp-book is also existent in english translation, and maybemaybemaybe even accessible online, don't know. (The german version of his book was/is accessible via digitizing-center of university of Göttingen, maybe this is perhaps a hint...)
- - - - -
To make the matter now a bit less prominent: having strong loss of energy since several monthes now I'm only skimming through my early elaborations and word-docs and if I find something worth to be looked at, which I did not present earlier here in the forum (but which is related to our matter), I think I'll add it "to the pipeline" for someone who might be interested by chance (or has a knack for historical matters)... Hmm, maybe a subforum for such thoughts might even be more appropriate...
Kind regards -
Gottfried
Gottfried Helms, Kassel
