Intereting Posts

Proper notation for distinct sets
Number of Trees with n Nodes
Simple binomial theorem proof: $\sum_{j=0}^{k} \binom{a+j}j = \binom{a+k+1}k$
taking the limit of $f(x)$, questions
Disjoint $AC$ equivalent to $AC$
What is this automorphism-related subgroup?
Show that this limit is positive,
An identity involving the Bessel function of the first kind $J_0$
How many ways can $8$ rings be put on $3$ fingers?
Limit of a sequence involving root of a factorial: $\lim_{n \to \infty} \frac{n}{ \sqrt {n!}}$
Find $\min(\operatorname{trace}(AA^T))$ for invertible $A_{n\times n}$
If $\frac{\sin^4 x}{a}+\frac{\cos^4 x}{b}=\frac{1}{a+b}$, then show that $\frac{\sin^6 x}{a^2}+\frac{\cos^6 x}{b^2}=\frac{1}{(a+b)^2}$
Error function etymology: Why the name?
Showing that a homomorphism between groups of units is surjective.
Issue with Spivak's Solution

I’ve seen recently for the first time in *Special Functions* (by G. Andrews, R. Askey and R. Roy) the definitions of fractional integral

$$(I_{\alpha }f)(x)=\frac{1}{\Gamma (\alpha )}\int_{a}^{x}(x-t)^{\alpha -1}f(t)dt\qquad \text{Re}\alpha >0$$

and fractional derivative

- Cauchy's residue theorem with an infinite number of poles
- Rolle's Theorem and the Mean Value Theorem
- composition of $L^{p}$ functions
- How to prove that the rank of a matrix is a lower semi-continuous function?
- Prove partial derivatives exist, but not all directional derivatives exists.
- Banach Measures: total, finitely-additive, isometry invariant extensions of Lebesgue Measure

$$\frac{d^{\nu }w^{\mu }}{dw^{\nu }}=\frac{\Gamma (\mu +1)}{\Gamma (\mu -\nu +1)}w^{\mu -\nu },$$

in *The Hypergeometric Functions* Chapter.

I would like to know some applications for Fractional Calculus and/or which results can only be obtained by it, if any.

- Can a non-periodic function have a Fourier series?
- Fourier transform of the Heaviside function
- True/False: Self-adjoint compact operator
- Is the max of two differentiable functions differentiable?
- Solve this integral:$\int_0^\infty\frac{\arctan x}{x(x^2+1)}\mathrm dx$
- Find $\lim\limits_{n \rightarrow \infty}\dfrac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}$
- Geometrical Interpertation of Cauchy's Mean Value Theorem
- Evaluation of $ \int_{0}^{1}\left(\sqrt{1-x^7}-\sqrt{1-x^4}\right)dx$
- Calculate in closed form $\sum_{n=1}^{\infty} \frac{\arctan(1/n) H_n}{n}$
- Integral of $\int \frac {\sqrt {x^2 - 4}}{x} dx$

Fractional derivatives can be used to establish connections between various special functions. The book An Atlas of Functions makes heavy use of this, especially derivatives of order 1/2 and -1/2.

Also, the *existence* of fractional derivatives is related to the convergence of Fourier transforms. For example, if a function has a 1/2 a derivative that means you can multiply its Fourier transform by $x^{1/2}$ and it is still in $L^2$. But I haven’t seen much use in actually computing fractional derivatives, only knowing that they exist.

On a somewhat related note, see my answer to a question on Math Overflow related to Sobolev spaces.

I wouldn’t say there are results that can only be obtained through differintegration. It only happens that there are problems whose solutions look neater when we bring in the machinery of differintegrals.

Spanier and Oldham and Miller and Ross remain useful references on the applications of differintegration. The first reference has a chapter on how certain diffusion problems have a neater formulation when differintegrals are used. For the second reference, the application that jumped out at me was Abel’s solution to the so-called tautochrone problem: finding the curve such that the time needed for a particle to descend from a given position to the bottom of the curve (assuming there is no friction) is *independent* of position.

Though Huygens and other mathematicians have already obtained this solution long before Abel, he decided to use an integral equation formulation that can then be solved with the help of differintegration. In particular, he arrived at the equation

$$\sqrt{2g}T=\int_0^y\frac{s^{\prime}(\eta)}{\sqrt{y-\eta}}\mathrm{d}\eta$$

which when reformulated as a differintegral is

$$\sqrt{\frac{2g}{\pi}}T=\frac{\mathrm{d}^{-\frac12}}{\mathrm{d}y^{-\frac12}}s^{\prime}(y)$$

I won’t spoil the rest of the solution; I’d suggest that you read Miller and Ross if you’re interested.

Miller and Ross looks very nice indeed.

As far as applications are concerned: Applications of Fractional Diﬀerential Equations. For some strange reason, the original link is not available. Instead, you can look at it by using Google Docs viewer.

Additional applications have arisen recently in fractional diffusion processes, mathematial biology (random eye movements follow a fractional process), solar physics, and many other places. In these cases the fractional derivatives are used (as noted above) primarily in writing models as fractional differential equations. A more detailed discussion of the fractional diffusion equation and its applications can be found here.

A short paper for general public was published on Scribd :

“The fractionnal derivation”

http://www.scribd.com/JJacquelin/documents

Concerning “which results can be obtained”, there are many ways of doing this, most of which involving complicated formulas or some other means.

The easiest method in my opinion is the simplistic induction. Take $D^n_x:=\frac{d^n}{dx^n}$ and it will be fairly obvious that $D^n_xe^{ax}=a^ne^{ax}$. One most easily proves this through induction, using the fact that $D^n_xD^k_x=D^{n+k}_x$ for $n\in\mathbb Q$, then $\mathbb R$ by assuming continuity.

Other formulas can be made, like generalizing the $n$th derivative through the limit definition of a derivative.

Some commonly known fractional derivatives include $D^n_xe^{ax}=a^ne^{ax}$, $D^n_x\sin(x)=\sin(x+\frac{n\pi}2)$, and $D^n_xx^\mu=\frac{\Gamma(\mu+1)}{\Gamma(\mu-n+1)}x^{\mu-n}$ if $-\mu\notin\mathbb N$.

Less known, for example, would be if $-\mu\in\mathbb N$, in which case, $D^n_x\frac1x=\frac{\ln(x)-\gamma-\psi^{(0)}(-n)}{x^{n+1}\Gamma(-n)}$. Obviously, the less commonly known fractional derivatives are much more complicated.

- Example of Left and Right Inverse Functions
- Can the graph of a bounded function ever have an unbounded derivative?
- Isometries of $\mathbb{R}^n$
- Expected value of two successive heads or tails (stuck on computation)
- $G$ is a group, prove: if $<∞$, then $$ is finite.
- Center of mass of an $n$-hemisphere
- $X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed
- Show that the set of functions $\mathbb{N}\to\{0,1\}$ is not countable
- Show that $A \setminus ( B \setminus C ) \equiv ( A \setminus B) \cup ( A \cap C )$
- Uniform convergence of geometric series
- How is $\mathbb R^2\setminus \mathbb Q^2$ path connected?
- Prove a lower bound for $\sum_{i=1}^n i^2$
- who first defined a tangent to a circle as a line meeting it only once?
- What is the norm of the operator $L((x_n)) \equiv \sum_{n=1}^\infty \frac{x_n}{\sqrt{n(n+1)}}$ on $\ell_2$?
- Finding XOR of all even numbers from n to m