So, I've made a tiny bit of progress. I still write it the way I write it instead of Loren's method, because I like the x=b=c=n=2 case which generates pi, and the easiest to find coefficients (a_m =(2m!)/2). Yup.. more #2. It's also a "circular" function in the sense that its points connect at k=infinity. It wraps around to x-x... or the zero point.
I am still curious about the general formula for coefficients for the series. I am focusing on the n=2 case, although I've discovered the 1st 2 coefficients for all cases, first 3 for n=2 and 3.
The n=2 case is specific.
There are 2 types of functions. An iterative forwards function (directly calculate as many radicals as one wishes):
k= number of radicals
b= \(\sqrt{2x}\)
c= x^2-x
\(\\
Q_{x,k} = b^k \, \times \, \sqrt{x - \sqrt {c+\sqrt {c +...}}} \\
\\\)
k=3 radicals:
\(
Q_{x,3} = b^3 \, \times \, \sqrt{x- \sqrt{c+\sqrt{c}}}\\\)
And a smooth "backwards" function, starting at k= infinity (which oscillates perfectly at the x=2 line for k<0, maybe because of distribution of primes? Or maybe because this is the only function like this where x=b=c=n. Loren??? It's like a zero point for this network of functions, creates double amplitude cosine oscillations (or 2 * cosine).
And I'd think someone lore rich already would know, and maybe not share it. Then mock me for thinking you wouldn't mock me. Holding this one bit of information out of my reach for months and months, when it's super simple.
\(k \, \to \, \infty \)
\(
\\
q_x =b^k \,\times\, \sqrt {x- \sqrt{c + \sqrt {c + \sqrt{c +...}}}}\)
at x=2, you get pi. Also, this is the only place that the function is balanced perfectly. x=b=c (=n too, but... we're doing the simple case in which we don't regard n)
Backwards function, which allows smooth iterative addition of radicals, which isn't the reason I want the coefficients- I want the oscillating functions for some artwork. So I can say I did some stuff. With the help of whoever helped me. I'm not bothering to define the coefficients for sine too, because there are a million sine varieties. The simplest case is the 2*cosine case.
\( {(\frac {Q_{x,k}}{b^k})}^2 = \frac {{q_x}^2}{a_1 \, b^{2k}} + \frac {{q_x}^4}{a_2 \, b^{4k}} -\frac {{q_x}^6}{a_3 \, b^{6k}}+ \frac {{q_x}^8}{a_4 \, b^{8k}} -\,\, ... = x- \sqrt{c+\sqrt{c+...} \)
Not to be circular, but you should be able to see how at k--> infinity, how you get the constants associated with specific x, if you put them into the function. Circular reasoning corner.
At k=0, you get \( \frac {{q_x}^2}{a_1 } - \frac {{q_x}^4}{a_2 } +\frac {{q_x}^6}{a_3 }- \frac {{q_x}^8}{a_4 } +\,\, ... = x^2 \)
At k=1, you get \( \frac {{q_x}^2}{a_1 \, b^2}-\frac {{q_x}^4}{a_2 \, b^4} +\frac {{q_x}^6}{a_3 \, b^6}- \frac {{q_x}^8}{a_4 \, b^8} +\,\, ... = x \)
At k=2, you get \( \frac {{q_x}^2}{a_1 \, b^4} - \frac {{q_x}^4}{a_2 \, b^8} +\frac {{q_x}^6}{a_3 \, b^{12}}- \frac {{q_x}^8}{a_4 \, b^{16}} +\,\, ... = x - sqrt{c}\)
Anyway. So I've found a general formula for the smooth "backwards" function, well, for some of its coefficients. I haven't found the general formula.
Not smart enough or experienced enough to get there yet, and it seems, from my experiences, that I'm going to be smooshed into slavery to worthless douchebags. So I'd like to learn this before that happens. Just so I can make something cool with it. Sort of a silly goal, considering the likely path to my eventual slavery implies I'll lose all knowledge of this. But I can't avoid the urge to try to make something good. meehhh... as long as it isn't for some specific rich douchebag, I don't feel horrible doing it.
Ok, so this connects infinite iterations of a function smoothly to single iterations. It allows one to extract, from the infinite iteration point, single iterations.
Not only that, pi exists at the x=b=c=n=2. Point. This mathematical coincidence is amazing.. it's a bit like Euler's thingy, but it has number 2 in it. And who does number 2 work 4? Fourier transforms, of coursier.
Pi, a cornerstone of all mathematics, exists at the leverage point of this function system. As does 2. There is a nice leverage point of #2s through all n, in which b= 2^(n/(n-1)) -2^(n-1). So n=3 has b= 2^(3/2) -2^(1/2)=2^(1/2). They actually make some nice simple cosine like series for all n (surprisingly simple expansions, easy to extract terms for). But the main leverage point is where all the 2s meet, and make pi. At n=2.
Which is crazy. e^i pi=-1? I see 2 2 2 2 2. Terrible twos.
I have yet to connect it to the Wallis product. Which is Pi/2. However, it definitely will have a connection to it as well, as pi/2 is the point at which the "sine" version of the leverage point of the function has coefficients 1/(2m-1)!. Ok, there you go. The connection to the Wallis product. It and the cosine point, are the only ones in which all coefficients are easily represented with factorials. Every other coefficient, for other n and x, ARE way more complicated.
So, I am just asking for someone who already has the answer, because me learning the answer the hard way is probably not going to happen, because I am on the fricken street, to tell me the general formula for coefficients.
Here they are:
a2: 2x (2x-1)
a4 : x (4x^2-1)
a6: 4x (8x^3-1) / (2x +5)
a8: x(16x^4-1) (2x+5) /(8x^2+6x+7)
a10: 2x (32x^5-1) (8x^2+6x+7) / (48x^4+64x^3+56x^2 +28x +21)
Do you know how I am extracting coefficients? The dumb hard way.
Code:
f5(x,n,k):=[b:abs(bfloat(x^n-x)),r:(b)^(1b0/n),
for i:2 thru k-1 do[
r:(r+b)^(1b0/n)],
r:(x-r),
pi2:r*(n*x^(n-1b0))^(k),
disp (pi2)
]$
kk:60;
nn:2b0;
xx:3b0;
f5(xx,nn,kk*6)$
z252:(pi2);
f5(xx,nn,kk)$
pi2:(pi2);
zzz:nn*xx^(nn-1b0);
a:z252-pi2;
a2:a*zzz^kk;
z252^2b0/a2;
factor(%);
b:z252^2b0/((3*5)/2);
b1:b-a2;
a3:b1*zzz^kk;
z252^3b0/a3;
factor(%);
b2:z252^3b0/((5^2*7)/2)-a3;
a4:b2*zzz^kk;
z252^4b0/a4;
factor(%);
b3:z252^4b0/((3^3*5^3*7*13)/(2^2*43))-a4;
/// some coefficient code:
cof1(x,n):=2*factor((n*x^n- x)/((n-1)));cof1(2,2);
cof2(x,n):=factor( (n*x^(n)-x)*3*(n*x^(n-1)+1)/ ((n-2)*(n*x^(n-1)+1)+n+1));
cof21(x):= factor(4*x^2-2*x);
cof22(x):= factor(2*x^2-x)*(2*x+1);
cof23(x):=factor((4*x*(8*x^3-1))/(2*x+5));
cof24(x):=factor(x*(16*x^4-1)*(2*x+5)/(8*x^2+6*x+7));
cof25(x):=factor((2*x*(32*x^5-1)*(8*x^2+6*x+7))/ (48*x^4+64*x^3+56*x^2 +28*x +21));