05/23/2014, 10:53 PM
Yes yes that is what I meant.
I anticipated your approximation for the zeros of the asymptotic function, in terms of Kneser's half exponential.
The Weierstrass product form does not only need a computation but also a proof ofcourse. Too illustrate why, I gave those counterintuitive functions as examples.
A vague possible conjecture could be that exp(g(z)) is the " integer growth rate " for " most " entire functions.
Im afraid to conjecture it , but the idea fascinates me and perhaps plays a role in a proof.
I remind you that 2 posts are still completely unadressed, although ofcourse you might not have the time or habit of doing more than one thing at a time.
As for your education , the fun thing is that " research is a hard - but open book - exam ". And you are certainly a respected member here.
Another intresting/counterintuitive remark about product forms ( weierstrass or other ) for entire functions is the fact that for instance exp(x) can be written in the form :
(1+c_n x^n)
valid for abs(x)<1.
That seems surprising and fascinating considering that exp(x) has a very different ( and dull ) weierstrass product form.
This can all be generalized like almost forever , but the point is to warn about expansions that might only work locally and are thus not the weierstrass expansion !
Another "counter-intuitive" result.
Btw first mentioned here by Gottfried and written about on his page :
" dream of a sequence "
link : http://go.helms-net.de/math/musings/drea...quence.pdf
- This also reminds me of the classic pentagonal theorem of euler ofcourse - which might turn into an argument against truncation ...
... and also continued fractions ... id better stop here.
But some day you will see continued fractions
regards
tommy1729
I anticipated your approximation for the zeros of the asymptotic function, in terms of Kneser's half exponential.
The Weierstrass product form does not only need a computation but also a proof ofcourse. Too illustrate why, I gave those counterintuitive functions as examples.
A vague possible conjecture could be that exp(g(z)) is the " integer growth rate " for " most " entire functions.
Im afraid to conjecture it , but the idea fascinates me and perhaps plays a role in a proof.
I remind you that 2 posts are still completely unadressed, although ofcourse you might not have the time or habit of doing more than one thing at a time.
As for your education , the fun thing is that " research is a hard - but open book - exam ". And you are certainly a respected member here.
Another intresting/counterintuitive remark about product forms ( weierstrass or other ) for entire functions is the fact that for instance exp(x) can be written in the form :
(1+c_n x^n)
valid for abs(x)<1.
That seems surprising and fascinating considering that exp(x) has a very different ( and dull ) weierstrass product form.
This can all be generalized like almost forever , but the point is to warn about expansions that might only work locally and are thus not the weierstrass expansion !
Another "counter-intuitive" result.
Btw first mentioned here by Gottfried and written about on his page :
" dream of a sequence "
link : http://go.helms-net.de/math/musings/drea...quence.pdf
- This also reminds me of the classic pentagonal theorem of euler ofcourse - which might turn into an argument against truncation ...
... and also continued fractions ... id better stop here.
But some day you will see continued fractions
regards
tommy1729
