〈 Berger | Dillon 〉

Well, I don't think you need fractional calculus at all, it just makes things unnecessarily complicated. I made the exact same animation simply by adding the next term in the Taylor series weighted by a number going from 0 to 1.

yes, you can interpolate any way you wish you can also make it by just replacing the n! with Γ(1+n) and summing over reals.. using the fractional derivative is more general it just happens that the fractional derivative of e^x is simple

What do you mean by "summing over reals"?

some evenly spaced sample like n=0 to n=20 in steps of 0.1

I see, so you mean applying trapezoidal rule to integrate the function x^t/Gamma(t+1). But this integral is only asymptotically approaching e^x for x going to infinity. For small x, you have no hope getting a good approximation. Have you really tried what you said?

Isn't the point of this animation to show how the approximation converges and gets better with more terms? It's plotting e^ix, and yes the initial frames of animation are very far from a unit circle and are thus a 'bad approximation' of e^ix, but I think that's missing the point.

You don't understand asymptotic analysis. As a phd in math, I will tell you that calculating e^ix the way the OP claimed to be done is NOT going to converge for small x, no matter how many terms you take, aka the claim was false and he didn't understand what he was talking about.