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+--- Thread: another infinite composition gaussian method clone (/showthread.php?tid=1693)
another infinite composition gaussian method clone - tommy1729 - 01/20/2023
another infinite composition gaussian method clone
Yeah nothing huge to mention.
But just another to add to the list of gaussian method , beta method and similar methods.
we want f(s+1) = exp( t(s) * f(s) )
and t(s) strictly rises from 0 at s = -oo to 1 at s = + oo in a fast way.
The gaussian method had t(s) = (1 + erf(s))/2.
The beta has t(s) = 1/(1 + exp(-s)).
the incomplete gamma method had t(s) = 1/(1 + inc.gamma(-s))
etc
You get the idea.
So we want to construct a t(s).
We need something that looks like erf(s) or tanh(s)
Let c = 2/pi
Then such a candidate is the " special function " :
c tt(x)
where tt(x) is the " tommy tan " function ( which I invented many decades ago as a teenager, and somewhat inspired me to do math ... on the other hand I forgot about it mainly lol )
see here where my friend mick asks about the value c :
https://math.stackexchange.com/questions/191008/a-curious-limit-for-frac-pi2
tt(x) = sum_(n>0) (-1)^[n+1] x^(2n-1) [(2n)! ln(2n)]^(-1)
or for the tex fans :
\[
tt(x) = \sum_{n>0} (-1)^{[n+1]} x^{2n-1} [(2n)! \ln(2n)]^{-1}\\
\]
( notice tt(-x) = - tt(x) )
then t(s) becomes (1 + c tt(s))/2.
NOTICE that c tt(s) is entire !
Now this function has been resurrected and given a life purpose , it might be nice to investigate its properties.
***
I have been thinking/dreaming about an addition function formula tt(a+b) = ... or an asymptotical addition formula.
But maybe that is just a dream not worth persuing.
***
( and ofcourse there are " fake function ideas " related to this but i have mentioned this already too often )
regards
tommy1729
RE: another infinite composition gaussian method clone - JmsNxn - 01/23/2023
Honestly, I'm super thankful that my ideas have affected you so much, Tommy. But I'd be much more interested in a discussion of the family of solutions.
Let's take:
\[
T[t](s+1) = e^{t(s) T(s)}\\
\]
Where the functions: \(t\) form a quasi space of functions such that:
\[
\sum_{j=1}^\infty |t(s-j)| < \infty\\
\]
And:
\[
T[t](s) = \Omega_{j=1}^\infty e^{t(s-j)z}\,\bullet z\\
\]
But the "addition" on this space is \(t_0 \oplus t_1 = \frac{t_0(s) + t_1(s)}{2}\), and \(t(\infty) = 1\). Both mine and your methods, and all of your methods belong to this space. I'm pretty fucking confident nothing is going to work for \(b=e\), it might for others. But base \(e\) is just so fucking volatile. And I'm confident both our methods fail at holomorphy in \(\mathbb{C}\).I'd be very surprised at this point if any \(t\) produces holomorphic functions. And if we do find one, it'll be partitioned as \(\Im s \to \pm \infty\) that \(t \to L^{\pm}\). Or something like that.... We can only find a representation for Kneser in this manner. We're not going to break things....
RE: another infinite composition gaussian method clone - tommy1729 - 01/24/2023
I am more optimistic