Re: dy/dt = ky^2 (Riccati) and its Solution

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 08/24/04 

Date: Tue, 24 Aug 2004 18:38:43 +0000 (UTC)

On 24 Aug 04 11:33:31 -0400 (EDT), Osher Doctorow wrote:
>The answer to the above question is that the Riccati differential
>equation has as solutions the generators t, 1/t of all functions
>that can be expanded in power series about 0 (MacLaurin Series) and
>their "closest approximation" to first order exp(t), but not
>necessarily the functions generated by those generators, where
>generation is via the sums of powers of the quantities t, 1/t, and
>Laurent series are included as (generalized) power series about 0.
>I'll try to discuss this further soon, except to repeat that exp(t)
>generates a wide variety of other functions, e.g., via exp(it) and
>by diagonal inversion log(t).

Exponentials are also of enormous importance in Lie groups and Lie
algebras and in complex analysis besides their importance in generat-
ing trigonometric functions. See "Exponentials form a basis of
discrete holomorphic functions," Christian Mercat (Technische U.
Berlin) arXiv:math-ph/0210016 8 Oct 2002. Mercat also proves that
on a convex set, the discrete polynomials also form a basis.

Osher Doctorow