From Osher Doctorow
Let's take a look at the power series expansion of y = exp(x):
1) exp(x) = sum x^n/n!, sum over n = 1 to infinity
Let Dn...n(f) be the nth derivative operator acting on a function f, in
which case we know that:
2) Dn...n(x^n) = n!
Therefore, we can write (1) as:
3) exp(x) = sum x^n/Dn...n(x^n)
Notice that the nth term here runs through n = 1, 2, 3, etc., so that
in practice it covers derivatives of all orders, but that the quotient
which is being added on the right hand side of (3) is a power function
divided by its nth derivative.
According to Garrett Birkhoff, as I have fairly often pointed out on
sci.physics, differential equations embody Causation, via the (time)
derivative. I call the derivative Birkhoff Causation. Although
Probable Influence/Causation (PI) is not Birkhoff Causation, it is
quite close to it as exemplified by the Riccati Differential equation
which describes PI and is arguably the simplest time derivative
equation (with linear/exponential and logistic subtypes).
Since (3) shows that the exponential function is the sum of (power)
functions divided by their nth derivatives which in turn are their "nth
Birkhoff Causations", this equation (3) is arguably the simplest power
series relating a function to its (nth Birkhoff) Causation.
Happily, Andrei Linde of Stanford in "Sinks in the landscape and the
invasion of Boltzmann brains," hep-th/0611043 v1 6 Nov 2006, makes his
most detailed argument up to date for the centrality of the exponential
function in high level physics (especially string theory). His 32
page paper is well worth reading .
Osher Doctorow
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