Question

Fractional calculus question part 2:

Answer 1

Apparently these two seem to be true:
\[\Large \frac{d^n}{dx^n}(e^x)=e^x\]\[\Large \frac{d^n}{dx^n}(x^p)=\frac{p!}{(p-n)!}x^{p-n}\]

However, if we look at the power series of e^x, then it would seem to mean that the 1/2 derivative of e^x evaluated at x=0 does not equal 1.

Answer 2

\[\Large e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}\] \[\Large \frac{d^{1/2}}{dx^{1/2}}(e^x)=\sum_{n=0}^{\infty}\frac{x^{n-1/2}}{(n-1/2)!}\]

http://www.wolframalpha.com/input/?i=(-1%2F2)!&t=crmtb01

And of course all other values of the gamma function are going to be positive.

So the first term will be 1/sqrt(x) so at x=0 it seems like e^x explodes. Why is this and what does it mean?