Are all fractional deriviatives/integrals of $e^x$ equal to $e^x$? I have learned through calculus that the derivatives and the indefinite integrals of the exponential function are the same (at integer arguments) but was wondering if this holds true for fractional derivatives/integrals, such as $\frac{1}{2}$, or do they turn into something monstrous?
I would test this in Wolfram Alpha but I cannot seem to get the entry correct for a fractional derivative/integral.
 A: Not monstruous, but not very simple. The fractional integrals/dérivatives of the exponential function involves the Incomplete Gamma function :
Page 10, section 6, in the paper "The fractionnal derivation"
http://www.scribd.com/JJacquelin/documents
A: In fact, the fractional integals/dérivatives are based on the Riemann-Liouville operator in which the lower bound (a) of the integral plays an important role : 
http://mathworld.wolfram.com/Riemann-LiouvilleOperator.html
The conventional  fractionnal calculus  states a=0 : 
http://mathworld.wolfram.com/FractionalCalculus.html
In this case, the fractional transform of the exponential function involves the Incomplete Gamma function as already said. 
But in the case of a=-infinity, corresponding to the Weyl's operator, the Incomplete Gamma term fades and the exponential term remains alone. So, with the non-conventional definition based on the Weyl's operator, the transform of the exponential function is a simple exponential function.  
A: This depends on the definition of fractional derivatives but the definition based on Newton series interpolation over positive derivatives gives all fractional derivatives to be the same, $e^x$:
$$f^{(s)}(x)=\sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
The same goes for the definition based on Fourier transform:
$$f^{(s)}(x)=\frac{i}{2\pi}\int_{-\infty}^{+\infty} e^{- i \omega x}\omega^s \int_{-\infty}^{+\infty}f(t)e^{i\omega t}dt \, d\omega$$
The both definitions coincide, and I consider them natural unlike other definitions that may produce a different result.
