Quantum Gravity 218.3: Riccati Differential Equation vs Generalized Exponential Function

From Osher Doctorow

The Generalized Exponential Function (GEF for short) has the formal
form for one argument x:

1) GEF(x) = sum ki x^i (sum from i = 0 to infinity)

where ki is either +/- 1/i! or 0 for i = 0, 1, 2, ...., and the ki are
allowed to be phase dependent or piecewise constant.

It generates exp(x), exp(-x), cosh(x), sinh(x), sin(x), cos(x), and I
am allowing x to be replaced by cx, x +/- c, and so on for constant
c.

The Riccati Differential Equation is:

2) dy/dt = A(t) + B(t)y + C(t)y^2

With t = x and the coefficients A(t), B(t), C(t) themselves either
arbitrary (or specified) or phase constants including 0, dy/dt of
equation (2) generates exp(kx), exp(-kx), cosh(kx), sinh(kx), and if
we allow complex solutions exp(ikx) for k real, it generates sin(kx),
cos(kx). It's typical to choose A(t), B(t), C(t) as either constant
(or in my case phase constants) or simple functions of time t like
simple polynomials or power functions (including negative powers) to
simplify things but also in my view because the restriction of (2) to
the second power or less of y makes the restriction of the
coefficients A(t), B(t), C(t) to the second power or less of t or
similar things more symmetric and more plausible. Once in a while
there may be a reason to choose more complicated functions for the
coefficients from the physics, but putting no restrictions on them
would seem to defeat the idea of simplification and brevity.

It is rather interesting arguably that the GEF and the "reasonable" or
"second degree or less coefficient" Riccati Differential Equation
yield similar solutions, although the Riccati DE seems to yield more
including rational functions of exponentials as in the Logistic
Differential Equation subcase.

Osher Doctorow





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