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I have been interested in fractional calculus for some time now, and I have seen "lots" of definitions of the $\frac {d^\alpha} {dx^\alpha}$ operator.

I started with the book The Fractional Calculus by Oldham and Spanier, and it comes as no surprise I favor Grünwald-Leitnikov derivative. It seems to me as a great definitions, becuse it directly generalizes the basic definiton of the derivative $\frac {df} {dx}=\lim_{h \rightarrow 0} \frac {f(x)-f(x-h)} {h}$. And it also produces the integral when $\alpha$ is set to be a negative number.

Another (which I think is the Liouville definition, but I'm not sure) generalizes the property of differentiating an exponential $\frac {d^k} {dx^k} e^{rx} = r^ke^{rx}$ and thus if $\frac {d^\alpha} {dx^\alpha}f(x)=\sum A_ne^{nx}$ then $f(x)=\sum A_n n^\alpha e^{nx}$.

A definition, which is used really often for some reason, is the Caputo derivative. Lot of people find it natural that $\frac {d^{\frac 1 2}} {dx^{\frac 1 2}} [1]=0$, but I think it is "evident" that it should be proportional to $x^{-\frac 1 2}$.

Now comes the actual question. Why are there so many definitions of the fractional derivative? Are some of them "better" than the others in some sense? And lastly, is there a general framework, wherein "functions" of differential operators, maybe more general than (fractional) powers, can be given explicit meaning?

I have been interested in fractional calculus for some time now, and I have seen "lots" of definitions of the $\frac {d^\alpha} {dx^\alpha}$ operator.

I started with the book The Fractional Calculus by Oldham and Spanier, and it comes as no surprise that I favor the Grünwald-Leitnikov derivative. It seems to me a great definition, because it directly generalizes the basic definition of the derivative $\frac {df} {dx}=\lim_{h \rightarrow 0} \frac {f(x)-f(x-h)} {h}$. And it also produces the integral when $\alpha$ is set to be a negative number.

Another (which I think is the Liouville definition, but I'm not sure) generalizes the property of differentiating an exponential $\frac {d^k} {dx^k} e^{rx} = r^ke^{rx}$ and thus if $\frac {d^\alpha} {dx^\alpha}f(x)=\sum A_ne^{nx}$ then $f(x)=\sum A_n n^\alpha e^{nx}$.

A definition, which is used really often for some reason, is the Caputo derivative. Lot of people find it natural that $\frac {d^{\frac 1 2}} {dx^{\frac 1 2}} [1]=0$, but I think it is "evident" that it should be proportional to $x^{-\frac 1 2}$.

Now comes the actual question. Why are there so many definitions of the fractional derivative? Are some of them "better" than the others in some sense? And lastly, is there a general framework, wherein "functions" of differential operators, maybe more general than (fractional) powers, can be given an explicit meaning?

I have been interested in fractional calculus for some time now, and I have seen "lots" of definitions of the $\frac {d^\alpha} {dx^\alpha}$ operator.

I started with the book The Fractional Calculus by Oldham and Spanier, and it comes as no surprise I favor Grünwald-Leitnikov derivative. It seems to me as a great definitions, becuse it directly generalizes the basic definiton of the derivative $\frac {df} {dx}=\lim_{h \rightarrow 0} \frac {f(x)-f(x-h)} {h}$. And it also produces the integral when $\alpha$ is set to be a negative number.

Another (which I think is the Liouville definition, but I'm not sure) generalizes the property of differentiating an exponential $\frac {d^k} {dx^k} e^{rx} = r^ke^{rx}$ and thus if $\frac {d^\alpha} {dx^\alpha}f(x)=\sum A_ne^{nx}$ then $f(x)=\sum A_n n^\alpha e^{nx}$.

A definition, which is used really often for some reason, is the Caputo derivative. Lot of people find it natural that $\frac {d^{\frac 1 2}} {dx^{\frac 1 2}} [1]=0$, but I think it is "evident" that it should be proportional to $x^{-\frac 1 2}$.

Now comes the actual question. Why are there so many definitions of the fractional derivative? Is some of them "better" than the others in some sense? And, at last, is there a general framework, wherein "functions" of differential operators, maybe more general than (fractional) powers, can be given explicit meaning?   

I have been interested in fractional calculus for some time now, and I have seen "lots" of definitions of the $\frac {d^\alpha} {dx^\alpha}$ operator.

I started with the book The Fractional Calculus by Oldham and Spanier, and it comes as no surprise I favor Grünwald-Leitnikov derivative. It seems to me as a great definitions, becuse it directly generalizes the basic definiton of the derivative $\frac {df} {dx}=\lim_{h \rightarrow 0} \frac {f(x)-f(x-h)} {h}$. And it also produces the integral when $\alpha$ is set to be a negative number.

Another (which I think is the Liouville definition, but I'm not sure) generalizes the property of differentiating an exponential $\frac {d^k} {dx^k} e^{rx} = r^ke^{rx}$ and thus if $\frac {d^\alpha} {dx^\alpha}f(x)=\sum A_ne^{nx}$ then $f(x)=\sum A_n n^\alpha e^{nx}$.

A definition, which is used really often for some reason, is the Caputo derivative. Lot of people find it natural that $\frac {d^{\frac 1 2}} {dx^{\frac 1 2}} [1]=0$, but I think it is "evident" that it should be proportional to $x^{-\frac 1 2}$.

Now comes the actual question. Why are there so many definitions of the fractional derivative? Are some of them "better" than the others in some sense? And lastly, is there a general framework, wherein "functions" of differential operators, maybe more general than (fractional) powers, can be given explicit meaning?

I have been interested in fractional calculus for some time now, and I have seen "lots" of definitions of the $\frac {d^\alpha} {dx^\alpha}$ operator.
I started with the book The Fractional Calculus by Oldham and Spanier, and it comes as no surprise I favor Grünwald-Leitnikov derivative. It seems to me as a great definitions, becuse it directly generalizes the basic definiton of the derivative $\frac {df} {dx}=\lim_{h \rightarrow 0} \frac {f(x)-f(x-h)} {h}$. And it also produces the integral when $\alpha$ is set to be a negative number.
Another (which I think is the Liouville definition, but I'm not sure) generalizes the property of differentiating an exponential $\frac {d^k} {dx^k} e^{rx} = r^ke^{rx}$ and thus if $\frac {d^\alpha} {dx^\alpha}f(x)=\sum A_ne^{nx}$ then $f(x)=\sum A_n n^\alpha e^{nx}$.
A definition, which is used really often for some reason, is the Caputo derivative. Lot of people find it natural that $\frac {d^{\frac 1 2}} {dx^{\frac 1 2}} [1]=0$, but I think it is "evident" that it should be proportional to $x^{-\frac 1 2}$.
Now comes the actual question. Why are there so many definitions of the fractional derivative? Is some of them "better" than the others in some sense? And, at last, is there a general framwork, wherein "functions" of differential operators, maybe more general than (fractional) powers, can be given explicit meaning?

I have been interested in fractional calculus for some time now, and I have seen "lots" of definitions of the $\frac {d^\alpha} {dx^\alpha}$ operator.

I started with the book The Fractional Calculus by Oldham and Spanier, and it comes as no surprise I favor Grünwald-Leitnikov derivative. It seems to me as a great definitions, becuse it directly generalizes the basic definiton of the derivative $\frac {df} {dx}=\lim_{h \rightarrow 0} \frac {f(x)-f(x-h)} {h}$. And it also produces the integral when $\alpha$ is set to be a negative number.

Another (which I think is the Liouville definition, but I'm not sure) generalizes the property of differentiating an exponential $\frac {d^k} {dx^k} e^{rx} = r^ke^{rx}$ and thus if $\frac {d^\alpha} {dx^\alpha}f(x)=\sum A_ne^{nx}$ then $f(x)=\sum A_n n^\alpha e^{nx}$.

A definition, which is used really often for some reason, is the Caputo derivative. Lot of people find it natural that $\frac {d^{\frac 1 2}} {dx^{\frac 1 2}} [1]=0$, but I think it is "evident" that it should be proportional to $x^{-\frac 1 2}$.

Now comes the actual question. Why are there so many definitions of the fractional derivative? Is some of them "better" than the others in some sense? And, at last, is there a general framework, wherein "functions" of differential operators, maybe more general than (fractional) powers, can be given explicit meaning?