One-Half Derivative: Solving Basics of Polynomials

[Feldoh]

A few days ago one of my teachers was talking about a half-derivative with basic polynomials and I got bored and tried to figure it out.

I found that we get
\frac{d^{a}y}{dx^{a}} x^n= \frac{(n!)x^{(n-a)}}{(n-a)!}

The problem is when you set a=1/2 I get (n-1/2)! in my denominator and I'm not really sure how to interpret that...

 

[LukeD]

Take a look at the Gamma function: http://en.wikipedia.org/wiki/Gamma_function

You should instead write it as

\frac{d^{a}y}{dx^{a}} x^n= \frac{\Gamma(n+1)x^{(n-a)}}{\Gamma(n-a+1)}

 

[Feldoh]

Thanks! However once again I seem to be stuck as to how so apply the gamma function.

Specifically this is mentioned:

(n+\frac{1}{2})! = \sqrt{\pi} \prod_{k=0}^n \frac{2k+1}{2}

But I'm not really sure how that is derived... Actually the gamma function itself just confuses me.

Also is there even any use to a half-derivative?

 

[Hurkyl]

Some buzzwords:

fractional derivative
fractional integral
fractional calculus