One-Half Derivative: Solving Basics of Polynomials

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Discussion Overview

The discussion revolves around the concept of half-derivatives in the context of basic polynomials, exploring the mathematical formulation and implications of fractional calculus, particularly the use of the Gamma function in this context.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • One participant presents a formula for the half-derivative of polynomials, leading to a factorial expression that raises questions about the interpretation of non-integer factorials.
  • Another participant suggests using the Gamma function to address the issue of factorials for non-integer values, proposing a revised formula involving Gamma functions.
  • A later reply expresses confusion regarding the application of the Gamma function and its derivation, particularly in relation to half-factorials.
  • Participants mention terms related to the broader field of fractional calculus, indicating a range of concepts that may be relevant to the discussion.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation and application of half-derivatives or the Gamma function, with ongoing confusion and questions remaining about these concepts.

Contextual Notes

There are limitations in understanding the derivation of the Gamma function's relation to half-factorials and the practical applications of half-derivatives, which remain unresolved in the discussion.

Who May Find This Useful

Readers interested in fractional calculus, mathematical analysis, and applications of derivatives in polynomial functions may find this discussion relevant.

Feldoh
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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
[tex]\frac{d^{a}y}{dx^{a}} x^n= \frac{(n!)x^{(n-a)}}{(n-a)!}[/tex]

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

Specifically this is mentioned:

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

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?
 
Last edited:
Some buzzwords:

fractional derivative
fractional integral
fractional calculus
 

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