# A general formula for the half-derivative?

1. Jul 15, 2011

### dimension10

$${H}^{2}x=\frac{dy}{dx}$$

Where

H is the Half-derivative operator.

My question is:

Is there a general solution for:

$$\frac{{d}^{\frac{1}{2}}}{{d}^{\frac{1}{2}}x}f(x)$$

or alternatively

$$H\; f(x)$$

I have another question. What is the meaning of the symbol:

$$\oint_{\alpha}^{\Omega}f'(x)\; dx$$

or

$$\oint f'(x)\; dx$$

I don't get what it means when you have that circle in the centre.

Thanks.

2. Jul 15, 2011

### tiny-tim

hi dimension10!

the phrase to google is "fractional derivatives" or "fractional calculus" (or just go straight to http://en.wikipedia.org/wiki/Fractional_calculus" [Broken] )

and ∫ on its own means between limits (which may be ±∞),

but ∫ with a circle means over a closed path or surface, ie without limits

Last edited by a moderator: May 5, 2017
3. Jul 15, 2011

### dimension10

But they have not specified a general formula

Thanks.

Last edited by a moderator: May 5, 2017
4. Jul 15, 2011

### tiny-tim

Last edited by a moderator: Apr 26, 2017
5. Jul 15, 2011

### dimension10

Last edited by a moderator: Apr 26, 2017
6. Jul 23, 2011

### dimension10

What is the notation for the opposite of the fractional derivative? As in,

$$\frac{d}{dx}f(x)=f'(x) \Rightarrow \int f'(x) dx =f (x)$$

$$\frac{{d}^{\frac{1}{2}}}{{dx}^{\frac{1}{2}}}f(x)=f'(x)\Rightarrow Statement(P)$$

Then what is the notation for the statement P?

Also, what is the opposite of the closed loop-integral thing?

Thanks.

7. Jul 23, 2011

### tiny-tim

sorry, don't know

8. Jul 23, 2011

### HallsofIvy

That same site tiny-tim links to gives another site:
http://en.wikipedia.org/wiki/Riemann–Liouville_integral
that gives
$$I^\alpha f(x)= \frac{1}{\gamma(\alpha)}\int_a^x f(t)(x- t)^{\alpha- 1} dt$$
as the "$\alpha$ integral" for any real $\alpha$.

The "closed loop integral" is like a definite integral- a number. It does not take one function to another and so has no inverse.

9. Jul 23, 2011

### dimension10

Thanks a lot.