Articles of fractional calculus

Some pecular fractional integrals/derivatives of the natural logarithm

(Be prepared for a very long post) I have deduced the following formula: $$D^{-n}\ln(x)=\frac{x^n(\ln(x)-n)}{(-n)!}=\frac{x^n(\ln(x)-n)}{\Gamma(-n+1)}$$ Where $$D^{-1}f(x)=\int f(x)dx$$ $$D^{-2}f(x)=\int\int f(x)dxdx$$ $etc$. $$D^0f(x)=f(x)$$ $$D^1f(x)=\frac d{dx}f(x)$$ $$D^nf(x)=\frac{d^n}{dx^n}f(x)$$ So the $n$th integral of $\ln(x)$ is given by my formula if $n$ is a natural number. Since the formula is continuous for $n\in\mathbb{R}$ when $n$ is not a negative integer, […]

Evaluating $\psi^{(1/2)}(x)$ of the extended polygamma function

The polygamma function is generally given by $$\psi^{(n)}(x)=\frac{d^{n+1}}{dx^{n+1}}\ln(\Gamma(x)),~n\in\mathbb N_{\ge0}$$ where $\Gamma$ is the gamma function. This can be extended to negative integers by letting $$\psi^{(n-1)}(x)=\int_a^x\psi^{(n)}(x)$$ Unfortunately, there is no fixed $a$ such that the above holds true for any $n$, but it does capture the general idea. Using fractional calculus, one can extend the polygamma […]

Fractal derivative of complex order and beyond

Is there some precise definition of “complex (fractal) order derivative” for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would like to know if some mathematician has defined a complex order derivative valid without restrictions for all complex number z. I mean: Is a well-defined definition […]

Fractional Order Derivative

A friend and I were talking about derivatives, and he asked an interesting question. Since both of us have not taken our calculus courses yet, neither of us were sure of the answer. His question has two parts: 1. can you have a fractional order derivative i.e. could you have $\frac{d^ny(x)}{dx^n}$ where, for example, $n=\frac{1}{2}$? […]

Taking the half-derivative of $e^x$

While attempting to teach myself the fractional calculus, I encountered a tragically early roadblock. For non-power rule fractional derivatives, I am having a lot of trouble evaluating for a closed form. Would someone mind walking me through the process for taking the half-derivative of $$f(x) = e^x$$ Really the most difficult part is evaluating $$\int_0^x […]

Calculating $\frac{\partial^{1/2}}{\partial x^{1/2}}\left( e^{-\alpha x^2 + \beta x} \right) $

I would like to calculate $$\frac{\partial^{1/2}}{\partial x^{1/2}}\left( e^{-\alpha x^2 + \beta x} \right) $$ My intuition is that I would have to use some sort of fractional Leibniz formula to first separate calculus of the half derivative of $e^{-\alpha x^2}$ from the other one, which is easy to derive $$ \frac{\partial^{1/2}}{\partial x^{1/2}}\left( e^{\beta x} \right) […]

What is the physical meaning of fractional calculus?

What is the physical meaning of the fractional integral and fractional derivative? And many researchers deal with the fractional boundary value problems, and what is the physical background? What is the applications of the fractional boundary value problem?

Half order derivative of $ {1 \over 1-x }$

I’m new to this “fractional derivative” concept and try, using wikipedia, to solve a problem with the half-derivative of the zeta at zero, in this instance with the help of the zeta’s Laurent-expansion. Part of this fiddling is now to find the half-derivative $$ {d^{1/2}\over dx^{1/2}}{1 \over 1-x}$$ First I would like to understand, whether […]

What does a “half derivative” mean?

I was looking at fractional calculus on Wikipedia, specifically this section and came across the half derivative of the function $y=x$ which is $y=\frac{2\sqrt{x}}{\sqrt{\pi}}$ . The derivative tells the slope at any point on the curve, but what does the “half derivative” mean – it’s obviously not $\frac{1}{2}$ the derivative of $y=x$ which would be […]

Do different methods of calculating fractional derivatives have to be equal?

Do different methods of calculating fractional derivatives have to be equal? Or do they sometimes end up differently? An example would be nice, and if possible, an explanation as too why such formulas can disagree with one another would be exceptional. The main reason behind this is because I noted that if we could take […]