Editing Fractional calculus (section)

===Riemann–Liouville fractional integral===
The classical form of fractional calculus is given by the [[Riemann–Liouville integral]], which is essentially what has been described above. The theory of fractional integration for [[periodic function]]s (therefore including the "boundary condition" of repeating after a period) is given by the [[Weyl integral]]. It is defined on [[Fourier series]], and requires the constant Fourier coefficient to vanish (thus, it applies to functions on the [[unit circle]] whose integrals evaluate to zero). The Riemann–Liouville integral exists in two forms, upper and lower. Considering the interval {{closed-closed|''a'',''b''}}, the integrals are defined as
<math display="block">\begin{align}
\sideset{_a}{_t^{-\alpha}}D f(t) &= \sideset{_a}{_t^\alpha}I f(t) \\
   &=\frac{1}{\Gamma(\alpha)}\int_a^t \left(t-\tau\right)^{\alpha-1} f(\tau) \, d\tau \\
\sideset{_t}{_b^{-\alpha}}D f(t) &= \sideset{_t}{_b^\alpha}I f(t) \\
  &=\frac{1}{\Gamma(\alpha)}\int_t^b \left(\tau-t\right)^{\alpha-1} f(\tau) \, d\tau
\end{align}</math>

Where the former is valid for {{math|''t'' > ''a''}} and the latter is valid for {{math|''t'' < ''b''}}.<ref>{{cite book |last=Hermann |first=Richard |date=2014 |title=Fractional Calculus: An Introduction for Physicists |edition=2nd |location=New Jersey |publisher=World Scientific Publishing |page=46 |isbn=978-981-4551-07-6 |doi=10.1142/8934 |bibcode=2014fcip.book.....H}}</ref> 

It has been suggested<ref name=Mainardi/> that the integral on the positive real axis (i.e. <math>a = 0</math>) would be more appropriately named the Abel–Riemann integral, on the basis of history of discovery and use, and in the same vein the integral over the entire real line be named Liouville–Weyl integral.

By contrast the [[Grünwald–Letnikov derivative]] starts with the derivative instead of the integral.