TITLE: Fractional derivative of exponential function
QUESTION [7 upvotes]: With the $n$th order derivative ($n$ as a positive integer) of $e^{ax}$ given by $$D^{n}e^{ax}=a^ne^{ax},$$
is the generalized (or fractional) derivative the same? Does it apply for any arbitrary $\alpha$? That is $$D^{\alpha}e^{ax}=a^{\alpha}e^{ax}?$$ For example...$$D^{\frac{1}{2}}e^{2x}=2^{0.5}e^{2x}?$$
Do you know of a study or paper showing how the integer order derivative of an exponential function is generalized into a fractional order?
Thank you.

REPLY [2 votes]: The short answer is: it depends. You have different definitions given different starting point $a$
http://mathworld.wolfram.com/Riemann-LiouvilleOperator.html
Look also here for a more extended discussion:
http://mathpages.com/home/kmath616/kmath616.htm