Quick question about fractional derivatives and is Latex broken?

[Kharakov]

Assume generalized binomial coefficients. Also assume this is not for normal fractional (iterative) derivatives, but for the following operation. I think I should implement it in python or something... just wonder if there is an easy way to do it on a phone (like how do I code it for Wolfram Alpha???) since my computer time is limited.

Latex broken? I specifically prohibit Bilby from replying to this part of the question if no forum user is wearing latex.

\(f^n(x) = \lim_{h \to 0} \frac{ \sum\limits_{k=0}^\infty \binom{\alpha}{k} (-1)^k f(x+ (n-k) h ) } {h^n}\)

So, limits as h-->0:
f⁰ (x) = [f(x) ]/h^0
f¹ (x) = [f(x+1h) - f(x) ]/h^1
f^1.5 (x) = [f(x+1.5h) - 1.5 f(x+.5h) + 3/8 f(x-.5h) +1/16 f(x-1.5h) +3/128 f(x-2.5h).... ]/h^1.5
f² (x) = [f(x+2h) -2 f(x+h) +f(x) ]/h^2
f³ (x) = [f(x+3h) -3 f(x+2h) +3 f(x+h) -f(x) ]/h^3

 

[barbos]

I don't think your formula work at all.

https://en.wikipedia.org/wiki/Fractional_calculus
Unlike integer, fractional derivative in general is not even local.

 

[steve_bank]

I am not familiar with the term fractional derivative, can you explain?

There are several free tools. Euler is a good one. They all have scripted languages to write code. They are all straightforward.

You can do it in a spreadsheet with Basic. Open Office is free and has basic with the spreadsheet.

 

[barbos]

I am not familiar with the term fractional derivative, can you explain?

Why ask when wikipedia exists?
https://en.wikipedia.org/wiki/Fractional_calculus

 

[steve_bank]

One other thing. Get a programmable TI or HP calculator. They have always had a serial, link to a PC. They may connect to a wireless device. Edit and download programs. Even without that you can program iterative solutions. Used them often before PCs.

 

[Kharakov]

I don't think your formula work at all.

Apparently it does. <shrug>

https://www.mathpages.com/home/kmath616/kmath616.htm

https://en.wikipedia.org/wiki/Fractional_calculus
Unlike integer, fractional derivative in general is not even local.

As far as I can tell, from what I just read (and it makes perfect sense to me), no derivative is local, because the derivative of any constant is 0.

Thanks barbos.

 

[Kharakov]

I am not familiar with the term fractional derivative, can you explain?

Try when I get back. Limited computer time.

lol. I still see your name as steve_bnk, without the a. I think of you as that too (I read "Steve Be En Kay"...).

I really like this article: https://www.mathpages.com/home/kmath616/kmath616.htm

 

[barbos]

I don't think your formula work at all.

Apparently it does. <shrug>

https://www.mathpages.com/home/kmath616/kmath616.htm

OK, I stand corrected. But I need to read it more. Easiest/most natural way to calculate or think of fractional derivative is through Fourier transformation.

https://en.wikipedia.org/wiki/Fractional_calculus
Unlike integer, fractional derivative in general is not even local.

As far as I can tell, from what I just read (and it makes perfect sense to me), no derivative is local, because the derivative of any constant is 0.

Thanks barbos.

Ordinary non-fractional derivatives are local: depend only of function behavior near argument. Fractional derivatives involves integration, so they depend on the boundaries and therefore not local.

 

[Kharakov]

So, I'm just basing this off of what I read in that article, which makes sense to me.

No derivative is local. The derivative of x^2 + 1000000000 = the derivative of x^2 - 159-8135907125309710298098510970912537097 = the derivative of x^2.

 

[barbos]

So, I'm just basing this off of what I read in that article, which makes sense to me.

No derivative is local. The derivative of x^2 + 1000000000 = the derivative of x^2 - 159-8135907125309710298098510970912537097 = the derivative of x^2.

.
That has nothing to do with locality

 

[Kharakov]

Let's take a simple example. f(x)=x + c f '(x)=1

How, is the derivative local, if any value of f(x) can have the same derivative? lol... this one is funny.

 

[barbos]

Let's take a simple example. f(x)=x + c f '(x)=1

How, is the derivative local, if any value of f(x) can have the same derivative? lol... this one is funny.

What is wrong with you?

 

[Kharakov]

Let's take a simple example. f(x)=x + c f '(x)=1

How, is the derivative local, if any value of f(x) can have the same derivative? lol... this one is funny.

What is wrong with you?

My sense of humor. Know its knot.

 

[lpetrich]

Finite-difference formulas are approximations for derivatives, not exact expressions. But there are exact formulas.

For integration, f^(-n)(x) = 1/((n-1)!) int of (x - t)^n-1 f(t) by t from t0 to x

For differentiation, f(x) = n!/(2*pi*i) contour int of f(t) / (t - x)ⁿ⁺¹ by t around x

One can use them to find fractional-derivative formulas, using the Euler gamma function to extend the factorial function to fractional args.

 

[RayJ]

Latex broken?

Latex is fixed!

 

[lpetrich]

Repeated integration:
\( f^{(-n)}(x) = \frac{1}{(n-1)!} \int_{t_0}^x (x - t)^{n-1} f(t) dt \)

Repeated differentiation:
\( f^{(n)}(x) = \frac{n!}{2\pi i} \oint \frac{f(z)}{(x-z)^{n+1}} dz \)

a contour integral, where the contour goes around x.

 

[Gun Nut]

...if no forum user is wearing latex...

I am wearing latex.

I always wear two condoms wherever I go. If I happen to find myself in a situation where I am about to have sex with someone, I remove one of the condoms... and boy, does it feel GREAT!

 

[Kharakov]

"I am wearing latex" was great to read, until I got to the part where you wear condoms. Thanks for the image. It will definitely help out in the future, when I don't want to hook up with a hot woman. Ok. It won't help in the future. I'm not going to have that chance. lol.

You're the first person I've spoke with who needed 2 condoms. Is it a genetic condition, or is one of them your gun?

I thought Inspector Sledgehammer was the only person who protected their gun during sex.

 

[Gun Nut]

"I am wearing latex" was great to read, until I got to the part where you wear condoms. Thanks for the image. It will definitely help out in the future, when I don't want to hook up with a hot woman. Ok. It won't help in the future. I'm not going to have that chance. lol.

You're the first person I've spoke with who needed 2 condoms. Is it a genetic condition, or is one of them your gun?

I thought Inspector Sledgehammer was the only person who protected their gun during sex.

no, the point is that with two condoms on, I can't feel a damn thing... so when I take one off, I'm SUPER sensitive and it feels great.

dumb joke originally intended to poke fun at those that say that they hate condoms because they numb the sensations.. which of course they do... So I say that (not only do I wear one...) I wear two!