Why is fractional integral called 'fractional'? The fractional integral operator $I_\gamma$ is defined by 
$$ I_{\gamma}f(x) = \int_{\mathbb{R}^n} \frac{f(y)}{|x-y|^{n-\gamma}}dy, \qquad 0<\gamma<n.$$
I want to know why it is called 'fractional integral' and how it was been thought.
 A: The "fractional integral" is called such because it generalizes the $n$-fold repeated application of the integration operator
$$[I^n] f = [\underbrace{I \circ I \circ \cdots \circ I}_{\mbox{$n$ copies of $I$}}] f$$
where the latter looks like
$$[\underbrace{I \circ I \circ \cdots \circ I}_{\mbox{$n$ copies of $I$}}] f = \left[x' \mapsto \underbrace{\int_{a}^{x'} \int_{a}^{x_{n-1}} \cdots \int_{a}^{x_1}}_{\mbox{$n$ instances of the integral sign}}\ f(x)\ dx\ dx_1 \cdots\ dx_{n-1}\right]$$
that is, repeatedly integrating the function $f$ with some given lower bound, here denoted $a$, i.e. taking the function, integrating it, then integrating the integral, then integrating the integral of the integral, and so forth, to the case where that the number of nested integrals, $n$, is no longer a whole number. E.g. you can take $\gamma = 0.5$ and that will give you "half an integration", such that two such half-integrations, done one after the other, will give you the same result as a single integration above. From the definition above, this wouldn't make sense since it is nonsense to say "half an instance of an integral sign" - or even better, what that means mathematically. (You could, perhaps, be literal and write a half of the sign, i.e. something that would look like a weird "j", but that is mathematically meaningless.)
The fractal integral can also be used to define a fractal differential, by differentiating the function either outside or inside of it - this gives two inequivalent extensions for differentiation known respectively as the Riemann-Liouville and Caputo fractal derivatives.
