Re: Quantum Gravity Via Expansion-Contraction 9.0: Surprising Relationships of Riccati y-y^2

From Osher Doctorow mdoctorow@xxxxxxxxxxx

Transseries are asymptotic combinations of powers, logarithms, and
exponentials provided they are finitely generated, and as Costin (13
Aug 2006 paper) points out, a large class of functions can be described
asymptotically in terms of them. Part of this is because they are
closed under quite a few common operations.

Costin gives an example of a transseries generated by x^(-1) and
exp(-x):

1) sum c_(km) exp(-kx)x^(-m)

where summation is from k, m = 0 to infinity and the coefficients
c_(km) (km is subscript) are complex.

Exponentials already assume considerable importance from Riccati
Differential equations including its subtypes exponential growth/decay
and Logistic Differential equations. Similarly for powers, at least
first and second powers. Logarithms are of course "diagonal"
(composition) inverses of exponentials at least for real logarithms.

Osher Doctorow

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