In the previous post, I wrote about a way to generalize derivatives and integrals into one function. What happens if a number other than an integer is passed to that function?
Here is the generalization from the last post:
Set k=2 and 
:
Now there’s a problem. Since a! is only defined for integers 0 or greater, a different way of calculating factorials is needed. Luckily, there exists a Gamma function defined for all real and complex numbers such that:
Substitute the Gamma function into the other equation and simplify:
So, the half derivative of 
is approximately equal to 
.
This can be used for ‘fractional integration’ as well, if a negative number is used for n.