So, I'm still fiddling around with this, but I have some good news!
So let's start by adding the following code:
And, throw away the rho function, as it isn't helping too much and tau is a bit simpler:
This allows us to cap at 210 iterations; which is giving pretty good accuracy. And now, the moment of truth:
The taylor series is converging well and fine in the left half plane. I'm having trouble at the moment translating this into a good push forward, so that we can get Abel_N(1+I,1) accurately. But if we restrict ourselves to way out in the left half plane, we can show the taylor series converging--rather well actually.
I'm going to keep working on this, but I think this is an "all is not lost" moment...
I think the key for accurate taylor series is to have large amount of iterations. This is inherent to pari as I see it. We can get accurate taylor series, but in order for large iterations, we need the value way off in the left half plane. Quite honestly, this results in a standoff. Can't do large iterations on the real line = Can't get accurate taylor series.
I think the code we need is to calculate \( w = e^s \) and for \( \Im(w) > 0 \) we have \( \lim_{w\to 0} \text{Abel}_N(w,1) = L \)--and focus on the iteration:
\(
\text{Abel}_N(\exp(1)*w,1) = \exp(\text{Abel}_N(w,1))\\
\)
And here is where I think theta mapping arguments will come in...
This means we have some function:
\(
\theta(z) = \sum_{k=-\infty}^\infty a_k e^{2 \pi i k z}\\
\)
In which,
\(
\lim_{\Im(z) \to \pi} \theta(z) = \infty\\
\)
I think in coding this; that Sheldon's perspective is the correct perspective. And since this decays to \( L \) at \( \Re(s) = -\infty \)--we can take advantage of Kneser. So finding a \( \theta \) such that:
\(
F_1(z) =\beta(z+x_0,1) + \tau(z+x_0,1)\\
= \text{tet}_K(z+\theta(z))\\
\)
I've also realized, if sheldon hasn't yet; that these taylor series are such horrible converging things because there are singularities (not just singularities, essential singularities) at \( \Im(s) = \pi \)...
At this point I have to admit that the taylor series I am calculating are the correct taylor series. Nowhere in the book of taylor series does it say that exp(func(-0.5)) - func(0.5) has to converge better than it does already. It's a good heuristic. But again, I'm of the opinion these taylor series converge very slow. We are just a hair's width away from an essential singularity. As you said yourself, for certain values of exp the taylor series isn't very accurate. This is a series of said points.
But I've managed to get my code a good amount better with 64 bit; let's leave it at that.
As far as I can go off of; these are wild taylor series with no nice decay conditions. But pointwise I'm getting perfect values. And additionally, locally converging taylorseries; which tells us local holomorphy.
I'm creating an update to my programming. It is necessary that the user use 64 bit pari. I've coded this much more elegantly. We get 100 digits point wise--and if you want series precision you have to move into the left half plane. This is to mean the taylor series only really converge in the left half plane.
As we push forwards with exponentials; all bets are off on the taylor series. But, they're correctly drawn from the iteration. They don't converge well because there are essential singularities near by. Within the radius of convergence are volatiles values.
I'm just finalizing the code at the moment, but I feel this is the best I can do.
So let's start by adding the following code:
Code:
Abel_N(z,y,{count=210}) = {
if(real(Const(z)) <= -200,
beta(z,y) + tau(z,y,count),
exp(Abel_N(z-1,y,count))
);
}And, throw away the rho function, as it isn't helping too much and tau is a bit simpler:
Code:
tau(z,y,{count=50}) ={
if(count>0,
count--;
log(1+tau(z+1,y,count)/beta(z+1,y)) - log(1+exp(-z*y)),
0
);
}This allows us to cap at 210 iterations; which is giving pretty good accuracy. And now, the moment of truth:
Code:
Y = Abel_N(-200+I+z,1)
%207 = (0.3181315052047641353126542516868002490601022446936799586401033709777181182840717528927210774601097169 + 1.337235701430689408901162143170855957855323110313139359786647162450775646763995803052118147567939595*I) + (5.54876673259671867090099471080669318106785322420394358343834194635267845596250827285933089578346 E-29 + 1.254276694929928406925790477592362989605855648696982726300442861383344851757185951068450692905412 E-28*I)*z + (-8.77610451137567890702516702107760213598080342427416618704660290578196459015232036838913486840284 E-29 + 2.65558294516824465594518189586093084041160292357429621843094754106936193317165658238042140999508 E-29*I)*z^2 + (6.70010662252486126722077043078401776251685624470141332630367309667524637043178003445335908533503 E-29 - 1.356518896593052873127419788268571846157069979539793387506208085249648334361660345141854465986784 E-28*I)*z^3 + (5.07187405891633435609385513945516330046880742308149045141520996847322745740004204937389514442906 E-28 + 1.601770513355758776698919019246208480708963464262029371220476223493976751742109641796248229042656 E-28*I)*z^4 + (5.02924589814571227481028487349480143201114449094763895454594929074623772587805969610280555501615 E-29 + 1.750662368216475754101406646648667325538950871544438396575589350687680745148590989900079179187547 E-27*I)*z^5 + (-5.21776781038142151238401826948176347767693901807438076479622075678235355108946788166412237108280 E-27 + 1.678116605195591639054171699539912173875772490077822266848014384423492171271409904601623538024171 E-27*I)*z^6 + (-8.26434343840890350722346904148768684811179570548850275022784345864054233872204418001701646759989 E-27 - 1.476874451000434105441105228997148244210708680526571004185624457222454938185134929871430315674361 E-26*I)*z^7 + (4.34754737664907610583429268212122186262617158825367860044457533991408458195502101243594732885252 E-26 - 3.19623139425425634643404792819225382327104820166019680421930982531248760617941078262654785955599 E-26*I)*z^8 + (1.211093080645117834552[+++]
func(x) = sum(j=0,34, polcoef(Y,j,z)*x^j)
%208 = (x)->sum(j=0,34,polcoef(Y,j,z)*x^j)
exp(func(-0.5)) - func(0.5)
%209 = 5.81195007554839531308913981535518319235468370597307177012228385948292519940500147967864485328014 E-23 + 3.358625715751082476775152384313478520427155759259773009359566195839925352925697325662647409142233 E-21*IThe taylor series is converging well and fine in the left half plane. I'm having trouble at the moment translating this into a good push forward, so that we can get Abel_N(1+I,1) accurately. But if we restrict ourselves to way out in the left half plane, we can show the taylor series converging--rather well actually.
I'm going to keep working on this, but I think this is an "all is not lost" moment...
I think the key for accurate taylor series is to have large amount of iterations. This is inherent to pari as I see it. We can get accurate taylor series, but in order for large iterations, we need the value way off in the left half plane. Quite honestly, this results in a standoff. Can't do large iterations on the real line = Can't get accurate taylor series.
I think the code we need is to calculate \( w = e^s \) and for \( \Im(w) > 0 \) we have \( \lim_{w\to 0} \text{Abel}_N(w,1) = L \)--and focus on the iteration:
\(
\text{Abel}_N(\exp(1)*w,1) = \exp(\text{Abel}_N(w,1))\\
\)
And here is where I think theta mapping arguments will come in...
This means we have some function:
\(
\theta(z) = \sum_{k=-\infty}^\infty a_k e^{2 \pi i k z}\\
\)
In which,
\(
\lim_{\Im(z) \to \pi} \theta(z) = \infty\\
\)
I think in coding this; that Sheldon's perspective is the correct perspective. And since this decays to \( L \) at \( \Re(s) = -\infty \)--we can take advantage of Kneser. So finding a \( \theta \) such that:
\(
F_1(z) =\beta(z+x_0,1) + \tau(z+x_0,1)\\
= \text{tet}_K(z+\theta(z))\\
\)
I've also realized, if sheldon hasn't yet; that these taylor series are such horrible converging things because there are singularities (not just singularities, essential singularities) at \( \Im(s) = \pi \)...
At this point I have to admit that the taylor series I am calculating are the correct taylor series. Nowhere in the book of taylor series does it say that exp(func(-0.5)) - func(0.5) has to converge better than it does already. It's a good heuristic. But again, I'm of the opinion these taylor series converge very slow. We are just a hair's width away from an essential singularity. As you said yourself, for certain values of exp the taylor series isn't very accurate. This is a series of said points.
But I've managed to get my code a good amount better with 64 bit; let's leave it at that.
As far as I can go off of; these are wild taylor series with no nice decay conditions. But pointwise I'm getting perfect values. And additionally, locally converging taylorseries; which tells us local holomorphy.
I'm creating an update to my programming. It is necessary that the user use 64 bit pari. I've coded this much more elegantly. We get 100 digits point wise--and if you want series precision you have to move into the left half plane. This is to mean the taylor series only really converge in the left half plane.
As we push forwards with exponentials; all bets are off on the taylor series. But, they're correctly drawn from the iteration. They don't converge well because there are essential singularities near by. Within the radius of convergence are volatiles values.
I'm just finalizing the code at the moment, but I feel this is the best I can do.