A real function $f(x),\;x>0$ is said to be in the space $C_{\mu},\;\mu\in\mathbb{R}$ if there exists a real number $p(>\mu)$ such that $f(x)=x^pf_1(x)$, where $f_1(x)\in C[0,\infty)$ and is said to be in the space $C^{m}_\mu$ iff $f^{(m)}\in C_\mu \;,\;m\in \mathbb{N}$.
R-L fractional integral: The Riemann-Liouville fractional integral operator of order $\alpha\geq0$, of a function $f\in C_{\mu}$, $\mu\geq-1$, is defined as $$J^{\alpha}f(x)=\dfrac{1}{\Gamma(\alpha)}\int_{0}^{x}(x-t)^{\alpha-1}f(t)dt\;\;, \alpha>0\;\;,x>0$$
The above definition is very popular and is often used by many authors in their papers.
I am having a doubt in the use of $\mu\geq-1$. Why it is taken ? It seems to me that it plays a role in showing convergence of series $\sum f(x)$. But I can't get it if I use some particular examples.
I am unable to visualize it properly. Can anyone help me out ? Thanks in advance.
