Calculus without limits

# Ultra Finite Evolution

The presentation from last Saturday features a symmetric version of the ultra finite algebra. The derivative is f(x+1/2)-f(x-1/2) and the basis for the polynomials are symmetric (x-m)(x-m+1)…(x+m). We have then an ultra finite Taylor theorem also in the complex which produces explicit data fitting solutions. There are some challenges in that we have to split up functions to even and odd functions. Also the explicit solution $exp(x+iy) = 2^x (1+i)^y$ which worked for d/dx f = f(x+1)-f(x) and d/dz = (d/dx – i d/dy)/2\$ satisfies no more d/dz exp(z) = exp(z) exactly. We can still write down solutions to i d/dz u = D u as a Taylor series exp(-iD z) u(0) but this does only remains bounded if we tilt the time axis in the complex plane z. What is appealing in this ultra finite time evolution is that otherwise difficult expressions are finite sums for every z. An example is $(1-z D)^{-1}$. An other difficulty is that f(z) = exp(a z) solving f'(z) = a f(z) is not the same than exp(w) with w=az. This relates to that $(1+a h)^{x/h}$ is not the same than $(1+h)^{ax/h}$ and that we have only the same thin in the limit h ->0. But in ultra finite calculus, we do not take limits.