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STP, STOP, I just figured something, it's all wrong, OMG, it's all WRONG !!!
Demonstration:
The coefficients of the power series solutions of certain non-linear
differential equations are generated by convolutions of the preceeding
coefficients. One example is the differential equation
x x'' + a (x')^2 = b (1)
Among the solutions of this equation (with appropriate choices of a,b)
are exp(t), sin(t), cos(t), (A+Bt)^n, A+Bt+Ct^2, and sqrt(A+Bt+Ct^2).
This last function represents the separation between any two objects
in unaccelerated motion. Other solutions include the cycloid relation
for (non-rotating) gravitational free-fall, and the radial distance
of a mass from a central point about which it revolves with constant
angular velocity and radial freedom.
The power series solution of equation (1) can be written
x(t) = c[0] + c[1] t + c[2] t^2 + c[3] t^3 + ...
where the coefficients c[i] satisfy the convolutions
n / b if n = 2
SUM A(k,n-k) c[k] c[n-k] = (
k=0 \ 0 if n > 2
with
A(k,j) = (a-1) j (k-j) + k(k-1)/2
Any choice of c[0], c[1], c[2], and c[3], with c[1]c[2] not zero,
determines the values of a,b and therefore all the remaining
coefficients. There are many interesting things about these sequences
of c[k] values. Focusing on just the sequences with |c[k]| = 1,
k=0,1,2,3, there are obviously 16 possible choices, but only 8 up to
a simple sign change. These 8 can be arranged as four groups of 2:
k I II III IV
--- -------- ---------- --------- ---------
0 1 1 -1 1 1 1 1 1
1 1 -1 1 1 1 -1 1 -1
2 1 1 1 -1 -1 -1 1 1
3 -1 1 1 1 -1 1 1 -1
4 1/2 1/2 3/2 -3/2 0 0 1 1
5 1/2 -1/2 5/2 5/2 4/5 -4/5 1 -1
6 -3/2 -3/2 9/2 -9/2 2/5 2/5 1 1
7 3/2 -3/2 19/2 19/2 -2/5 2/5 1 -1
8 3/8 3/8 133/8 -133/8 -1/2 -1/2 1 1
9 -29/8 29/8 267/8 267/8 1/30 -1/30 1 -1
etc
Yhe coefficients in each group differ only in sign. The coefficients
in groups I and II diverge, and those in group IV are all units. Only
the group III sequences converge. Interestingly, these coefficients
are given very closely by
c[k-1] = 2 exp(ku) sin(kw)
for k>2, where
u = -0.145370157...
/ 1.877672951... for III(a)
w = (
\ 1.263919649... for III(b)
Notice that the two possible values of w sum to 3.1415926...
The integer numerators and denominators of these c[k] sequences also
have many interesting properties. For example, primes p congruent to
+1 (mod 4) first appear in the denominator at c[p], whereas primes
congruent to -1 (mod 4) first appear at c[p^2]. The sequence of
numerators is much less regular
1 -1 -1 1 0 -4 2 2 -1 -1 59 -9 -1 233 8 -934 49 .. etc
Incidentally, the value of b in the ubiquitous equation (1) is
essentially just a constant of integration, and the underlying
relation is the derivitive
x x'' + q x' x'' = 0
where q=3 for unaccelerated separations and q=2 for (non-rotating)
gravitational separations. Isolating q and differentiating again
leads to the basic relation, free of arbitrary constants,
x x' x'' x'''' - x x' (x'')^2 - x (x'')^2 x''' + (x')^2 x'' x''' = 0
Dividing by x x' x'' x''' gives the nice form
x'''' x''' x'' x'
------ - ----- - ---- + ----- = 0
x''' x'' x' x
see what i mean???
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