FTC states the existence of the integral;

Integral[f(x)dx,a,b] = F(b)-F(a)

An iteration of FTC gives Taylor's expansion and maps e^z to polynomials;

e^z = Sum[z^n/n!, n=0,...,Infinity]

The weighing factor 1/n! diminishes the higher polynomials and ensures absolute convergence.

It reminds the number of states of ESet;

SEn = n!(n-1)!(n-2)!...1!0!

ESet is derived from iterating a simple operation, like the e^z sum.

Then one would expect the Integral[] definition to be like the ESet operator E.

ESet accumulates its past and determined uniquely by its initial and final states, like Integral[].

ESet seems isomorphic to e^z.

## Sunday, March 26, 2006

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