Questions on the differentiation/integration of functions to arbitrary order.

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Fractional order Riemann Stielts integral

The definition of fractional order integral is well-known. Is there any definition for fractional order Riemann Stieltjes integral?
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2answers
319 views

Can the following trick be expanded upon?

Main Question What is the expansion of $d^{1+\epsilon}?$ Background I noticed the following trick (sometimes more laborious) to directly differentiate $ f(x) $ twice without differentiating it even ...
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30 views

Is the Riemann–Liouville fractional derivative holomorphic in order?

If my understanding of complex analysis is correct then the arbitrary order generalization of Cauchy's formula for repeated integration $$(J^\alpha f) ( x ) = { 1 \over \Gamma ( \alpha ) } \int_0^x ...
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1answer
40 views

Non-integer order derivative

I do not know much about fractional calculus, except what I have read in a few short posts at MSE and https://en.wikipedia.org/wiki/Fractional_calculus. I know that order of a derivative can be ...
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1answer
44 views

Example of a function that has fractional derivatives of order less than 1 but not 1

I have recently learned that some fractals can have fractional derivatives of order less than 1, say of 1/2 even if they are not differentiable (have no derivative of order 1). I wonder if there is a ...
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41 views

Generating fractional taylor series

I was considering the notion of taylor series which posit that the sum $$ \sum_{i=0}^{\infty} \frac{1}{i!} a_ix^i $$ Where: $$ a_i = \frac{d^if}{dx^i}_{x= a} $$ Converge to the function f in a ...
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25 views

Geometric and physical significance of differintegrals

Consider a fractional integral or derivative, what can we associate to them in geometrical or physical terms, I have seen that for fractional derivatives the greater the order of derivative the less ...
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4answers
157 views

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 ...
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1answer
34 views

What is the integral of $e^{a\cdot x+b\cdot y}$ evaluated _under_ the Koch Curve

Grew out of frustration about this question; just replace "over" by "under": What is the integral of $e^{a\cdot x+b\cdot y}$ evaluated over the Koch Curve What is $$ \iint_{K} e^{a \cdot x + b ...
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1answer
39 views

References for the operator $(I-\Delta)^{\alpha /2}$

I am studying PDEs involving fractional differential operators, and I have found a few properties for the operator $(I-\Delta)^{\alpha /2}$ scattered through scientific papers. I wonder if there is a ...
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1answer
166 views

Does Fractional Calculus define the derivative over the Weirstrass Function?

I recently read this paper on defining the fractional derivative for the Wierstrass function. This seems very interesting since derivatives over fractals are generally not well defined. Yet, this ...
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1answer
44 views

Relation between Laplace and Fourier transform

I have a function that has the property $\tilde f(s) = \tilde{f}(abs(s))$. For this function, I need the inverse Fourier transform. I actually know the inverse Laplace transform of $\tilde f$ and I ...
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1answer
29 views

Derivative of a function which is defined as a derivative

I'm new to this kind of stuff so maybe this is a stupid question but I don't even know what to search on the internet. My problem is that: find the derivative of the following function on $\Bbb R^3$ ...
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2answers
65 views

Geometric Interpretation of Fractional Derivatives

I was looking for a geometrical interpretations of fractional derivatives and fractional integrals. I would be glad to see any kind of intuitive and preferably visual interpretation of the objects ...
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0answers
72 views

Does Fractional Calculus have a real connection with Fractals? (or is it just an extra variable trick)

The fractional derivative and integral (operators that let you differentiate or integrate a fractional number of times) have drawn a lot of attention from people outside the field. Yet, after reading ...
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0answers
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Fractional-order Halanay-type inequality

If there is any research article or books are available for Fractional-order Halanay-type inequality, if yes means, Please provide the proof or references list.
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The boundness of the solution of a fractional differential equation

D is the notation of Riemann-Liouville derivative. When I did the common first order differential equation and prove the same thing, I am able to first solve it and then prove the statement. But ...
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53 views

Fractional derivative definition

Suppose that $f(x) \in C^1$ for a $x \in [a, x]$. Then a regularization of Riemann-Louisville fractional derivative is defined as: $ \frac{1}{\Gamma(1-b)} \frac{d}{dx} \int_{a}^{x}\left( ...
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Understanding fractional-differencing formula

I have a time series $y_t$ and I would like to model it as an ARFIMA (a.k.a. FARIMA) process. If $y_t$ is integrated of (fractional) order $d$, I would like to fractionally-difference it to make it ...
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43 views

Show that a complex number's set is no empty

Consider $\alpha \in \mathbb{C}$ such that $Re (\alpha) > |\alpha|^2.$ Why is the set $$\Omega_{\alpha}=\mathbb{C}^{*} - \{\lambda^{\alpha}e^{i\theta\alpha}; \lambda > 0 \ \mbox{and}\ -\pi \leq ...
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1answer
55 views

Integral over Fractals with respect to fractal dimension

I understand that there is type of integral with respect to measures that can return values when evaluated over an integral. But is there an Integral that returns d dimensional volume where d is the ...
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0answers
20 views

fractional system fractional order equations

As resolved and how is the graph the phase plane of $D^{\alpha}y_{1}(t)=2y_{1}(1-\dfrac{y_{1}}{2y_{2}}-\dfrac{y_{1}}{2})$ $D^{\alpha}y_{2}(t)=3y_{2}(1-\dfrac{y_{2}}{2y_{1}}-\dfrac{y_{2}}{2})$ ...
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1answer
44 views

What can I do with half-derivative?

If I want to know the slope of the tangent of curve function I just have to find the derivative of this function and if I want to know the area under this curve I can integrate it (the function of ...
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35 views

properties of fractional calculus using grunwald letnikov

First definitions of fractional integral and derivative I 'm using $I^{\alpha}f(x)=\displaystyle\frac{1}{\Gamma(\alpha)}\displaystyle\int_{0}^{x}(x-t)^{\alpha-1}f(t)dt$ y ...
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2answers
102 views

Imaginary-Order Derivative

I would like to find the imaginary-order derivative of a function (let's just focus on a simple function for now). There is the Riemann-Liouville fractional-derivative: $$ _{a}D^{i}_{t} f(t) = ...
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63 views

Computing the fractional derivative of a fractional integral

I know that $D^{\alpha}I^{\alpha}f(x)=f(x)$ and $D^{\alpha}I^{\beta}f(x)=D^{\alpha-\beta}f(x)$ but How can prove this? ...
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Is $(-\Delta)^{s}$ c0incident with $(-\Delta)^{s/2}$?

We already know the following facts: $$\displaystyle (-\Delta)^su(x):=c_{n,s}\text{P.V.}\int_{\mathbb{R}^N}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,dy, $$ where $s\in (0,1)$. $$\int_{\mathbb{R}^N} ...
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2answers
47 views

Is the Fractional integral operator well-defined?

How to prove the fractional integral operator $J_{\alpha}:L^p(\Bbb R^+)\rightarrow L^p(\Bbb R^+)$ (of order $\alpha>0$) which is defined for each $f\in L^p(\Bbb R^+)$ by $$J_{\alpha}f(x):={1\over ...
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51 views

Can a Local Fractional Differential Operator exist?

Let $f: \mathbb{R} \rightarrow \mathbb{R}$. The derivative of $f$ is defined pointwise, and we say that $f$ is differentiable if the derivative exists in each point. Higher order derivatives are ...
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1answer
55 views

Formula for tangent derivatives, how to prove?

How to prove? $$(\tan x)^{(s-1)}=\pi^{-s}\Gamma(s)\left(\zeta\left(s, \frac12-\frac x\pi\right)+(-1)^s\zeta\left(s, \frac12+\frac x\pi\right)\right) $$
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1answer
115 views

What level of math is needed to learn fractional calculus?

I was skimming through wikipedia pages and stumbled upon the fractional calculus page. My interest increased when I noticed it has applications in physics. I was wondering as an undergraduate who's ...
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1answer
60 views

Is fractional order Sobolev spaces reflexive?

Let $0<s<1$, we define $$ W^{s,p}(\Omega):=\left\{u\in L^p(\Omega),\,\frac{|u(x)-u(y)|}{|x-y|^{\frac{N}{p}+s}}\in L^p(\Omega\times\Omega)\right\} $$ with norm $$ \|u\|:=\left(\int_{\Omega} ...
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Solutions of fractional linear dynamical systems

The Mittag-Leffler function is defined as: $$ E_\alpha(\tau) = \sum_{k=0}^{\infty}\frac{\tau^k}{\Gamma(\alpha k + 1)}, $$ which can also be defined, analogously, for matrices $A\in\mathbb{R}^{n\times ...
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238 views

Is this similarity just a coincidence?

Here is the function $-1/x$: If we add infinitely many similar functions with a shift of pi/2 each in both directions, we get $\tan x$. But if we do the same only in one direction, we get ...
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2answers
68 views

n-th Derivative

It can be proven the for a function $h(x)=f(x)g(x)$, letting $f^{(k)}(x)=\frac{d^k}{dx^k}f(x)$ and $g^{(k)}(x)=\frac{d^k}{dx^k}g(x)$ then the n-th derivative, for n is an integer is: ...
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1answer
32 views

Simplifying general formula for fractional derivative by removed derivative of integral.

On the wikipage about fractional calculus, there's a general formula for the fractional derivative: $D^\alpha$ is the derivative operator. $$D^\alpha ...
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1answer
61 views

Do smooth functions have fractional derivatives of all orders?

Suppose $\nu > 0$ and $n$ is such that $\lceil\nu\rceil = n$. The Riemann-Liouville definition of the fractional derivative would be $$f^{(\nu)}(x) = ...
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Geometric mean of fractional derivative

What is the geometrical mean of the fractional derivative (of order $\alpha \in (0,1)$) for a function $f:\Bbb R \rightarrow \Bbb R$? For example $f$ is increasing on $\Bbb R$ if $f'$ is positive.
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1answer
42 views

fractional derivative of test function

I have a small questions concerning the fractional derivative of a test function. Is it true that if $u \in C^{\infty}_{c}(\mathbb{R})$ and we define the fractional derivative of this function as ...
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3answers
80 views

Non integer derivative of $1/p(x)$

I need to find the $k$'th derivative of $1/p(x)$, where $p(x)$ is a polynomial and $k\in\mathbb{R}$ It dosen't have to be an explicit formula, an algorithm which finds a formula for some $k$ is fine. ...
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1answer
73 views

Fractional powers of the operator $B: L_2(\mathbb{R}) \mapsto L_2(\mathbb{R})$, $Bf = f-f^{''}$.

Consider the linear operator $B: L_2(\mathbb{R}) \mapsto L_2(\mathbb{R})$ defined by the following mapping: $Bf = f-f^{\prime\prime} \equiv (I-\Delta)f$, where $\Delta$ is the Laplace operator that ...
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1answer
48 views

Fractional Derivatives on a function with bounded Support

I have a question about functions that have bounded support in $\mathbb{R}$. In particular, suppose that I have a function $f$ with support $A\subset \mathbb{R}$ so that $A$ is compact. Without loss ...
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76 views

Fractional derivatives of Gamma function

For integer $n \geq 0$, we have $\dfrac{d^n}{ds^n} x^s = (\ln x)^n \,x^s$. From this it follows, for example, that $$\int_0^{\infty}e^{-x}\ln^n x \,dx= \Gamma^{(n)}(0)$$ Question: is there a way of ...
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Fractional Derivatives

If we define the (forward) difference operator as $$\Delta f(x)=f(x+\Delta x)-f(x)$$ we can break it up using the "shift" operator $E\,f(x)=f(x+\Delta x)$ and the "identity" $1\,f(x)=f(x)$. Then ...
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1answer
154 views

Definition of fractional Laplacian on a compact manifold?

How does one define the fractional Laplacian operator $(-\Delta)^s$ on a compact Riemannian manifold? In $\mathbb{R}^n$, it is defined $$ (-\Delta)^s f(x) = c_{n,s} \int_{\mathbb{R}^n} \frac{f(x) - ...
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41 views

fractional derivaitve of logarithm function $x^ {a} log(x) $

Given the function $ x^{a}\log(x) $ natural logarithmic Could someone tell me how to evaluate the fractional derivative $$ \frac{d^{b}}{dx^{b}}x^{a}\log(x) $$ for positive $a$ and $b$
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4answers
801 views

Fractional Calculus: Motivation and Foundations.

If this is too broad, I apologise; let's keep it focused on the basics if necessary. What's the motivation and the rigorous foundations behind fractional calculus? It seems very weird & ...
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How should I interpret this function notation?

I'm trying to implement an FDGD Algorithm from a paper and I'm a little stuck how to interpret a piece of function notation. See page 7, equations 2 and 3 in this document: In there we have ...
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37 views

Convergence of a limit

everyone. I have a question regarding the convergence of a certain limit. I've been fiddling with it but its been proving quite evasive. What I am trying to calculate is the Grundwald-Letnikov ...
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1answer
145 views

Do fractional derivatives maintain the $[fg]'=f'g+g'f$ and $f(g(x))'=f'(g(x))\cdot g'(x)$ rules?

Of course, I'm not really familiar with all fractional derivative methods, but is it a necessary rule that they all should comply with? If not, which ones, for example, do and which don't ? ( ...