# Questions tagged [fractional-calculus]

Questions on the differentiation/integration of functions to arbitrary order. Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation and integration operators.

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### How to find half derivative of $x^{-\frac{1}{2}}$?

I use this general definition to do fractional differentiation: $$(D^nf)(t)=\frac{1}{\Gamma(1-n)}\frac{d}{dx}\int_a^x (x-t)^{-n}\space f(t)\space\space dt,\space\space 0<n<1$$ However, when I ...
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### What is the generalization of Taylor's expansion with the fractional number as a power index?

Given a classical Taylor expansion as $f(x)=\sum_{i=0}^{\infty} \frac{d^{i}f(x_{0})}{dx^{i}}\frac{(x-x_{0})^{i}}{i!}$ where i is a nonnegative integer. Can we generalize this expansion to fractional ...
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### Differentiating using Cauchy's repeated integral formula [duplicate]

Can we use Cauchy's repeated integral formula to differentiate functions? $$(I^nf)(t)=\frac{1}{\Gamma(n)}\int_a^x (x-t)^{n-1}\space f(t)\space\space dt,\space\space n\notin \mathbb Z_{\leqslant 0}$$ ...
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### Calculating half derivative of $x$ using Cauchy's repeated integral formula

As far as I know, Cauchy's repeated integral formula is defined as: $$(I^nf)(t)=\frac{1}{\Gamma(n)}\int_a^x (x-t)^{n-1}\space f(t)\space\space dt,\space\space n\notin Z_{\le 0}$$ I know that Gamma ...
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### Solution to “Heat Equation” with Fractional Laplacian in 2 Dimensions

Statement of the Problem We consider the equation: $\partial_t u + (- \Delta)^{1/2}u = 0$ for $u : \mathbb{R}^2 \rightarrow \mathbb{R}$. I would like to find a non-trivial solution to this ...
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### How to use Pi and Gamma functions to find factorial of a number? [duplicate]

How can I find $\pi!$ or $e!$ or $(4-\pi)!$ etc. I have done it for $(\frac{3}2)!$ but Gaussian integral was used and it was easy but when I try for such values, it is very hard.
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### N-th derivative of n-fold integral

I want to justify that the n-th derivative of an n-fold integral gives the original function. In other words that $$\frac{d^n}{dx^n}\frac{1}{(n-1)!}\int_{0}^{x}(x-s)^{n-1}f(s)ds=f(x)$$ If I ...
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### Derivation of $(iw)^{n}$ into the format $A+ib$

How to derive $(iw)^{n}$ into the format $A+ib$ ($A$ is the real part, $B$ is the imaginary part), where $i=\sqrt{-1}$, $w<0$, $n$ is a fractional number such as $\frac{1}{2}, \frac{2}{3}, ...$? ...