From Osher Doctorow
The Laurent Series is key in complex analysis, but the truncated real
Laurent series, truncated at quadratic powers, will be of interest
here. Let's write this as:
1) L(r) = k1r^2 + k2r + k3 + k4/r + k5/r^2
Choose k1 = 1, k2 = -1, k3 = -1 for our example, and let k4 and k5 be
positive real numbers. Then, writing L1(r) as the particular case of
L(r) for these constants, we have:
2) L1(r) = r^2 - r - 1 + k4/r + k5/r^2
Notice that asymptotically this approaches r^2 - r - 1, the Golden
Mean function (which when equated to 0 yields the Golden Mean). It
is in fact an approximation to the Golden Mean for large r.
But in the previous posting, we obtained a similar result by using as
the only nonzero coefficients of (1) the constants k3, k4, and k5. So
(1) has a "double" approach to the Golden Mean.
But why truncate the Laurent Series at the second (positive and
negaive) power? Why not? After all, Einstein truncated his equation
at the second (covariant) derivative and obtained General Relativity.
Here we obtain a force equation of Newtonian gravitation and an
outward expansion force asymptotically related to the Golden Mean with
a certain choice of constants.
Osher Doctorow
.
|
