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Okay so we know that \(f'(x)\) is the first derivative of \(f(x)\), which can be also expressed as \(f^{(1)}(x)\). And we know that if we take another derivative, it'll become \(f^{(2)}(x)\), or \(f^{(1+1)}(x)\).

When we integrate, we ask for the function whose derivative is the integral's argument function. So \(\int{f^{(2)}(x)dx}\) is \(f^{(1)}(x)\), or \(f^{(2-1)}(x)\) (let's ignore \(Cs\) for now).

Thus, we can say that \(f^{(-1)}(x)\) is the antiderivative of \(f(x)\), or its integral. With this in mind,

What is \(f^{(0.5)}(x)\)?

And how about \(f^{(i)}(x)\)?

Okay, so let's see. Well first of all, what would that even look like? Would you draw half a dash for the prime above \(f(x)\)? Would you draw half an integral for \(f^{(-0.5)}(x)\)? Which half, the top half or the bottom half? Well, I think that's the least of the issues.

Here's what I propose:

If we're to stick to basic calculus and only use derivatives and antiderivatives, then, in other words, we're only using integers. So we're trying to get a rational number by using only integer sum and difference. How are we going to do that? We're going to cheat:

Consider the following:

\(\int{\frac{d}{dx}\int{\frac{d}{dx}\int{\frac{d}{dx}}}} ... \int{\frac{d}{dx}}f(x)dx ... dxdxdx\)

\(\Rightarrow f^{\left( \displaystyle \sum_{n=1}^{\infty} {(-1)^n} \right)}(x) = f^{(1-1+1-1+...)}(x)=f^{(0.5)}(x)\).

And yet, I still don't know what \(f^{(0.5)}(x)\) would be (because I can't perform an infinite differentigration). What would it even mean?

Oh, and good luck on \(f^{(i)}(x)\).


\[\left| x \right|<0\] \[\Rightarrow x\in ?\] \[\nearrow\] [I'll tell you what the question mark is.]

UPDATE: The issue is resolved! View Fractional Calculus

Link Idea Credit: Michael Mendrin

Link 2: Differintegral (what I initially named this note, but then thought Derintegral was more brief. Should've left it ;))

Check out the cool concept application of Derintegrals:



Guess what this is?

Note by John Muradeli
2 years ago

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I think the question mark should be \(\phi\) , an empty set, isn't it?! Hasan Kassim · 2 years ago

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@Hasan Kassim Uum... John Muradeli · 2 years ago

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@Hasan Kassim O_O John Muradeli · 2 years ago

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This is called "fractional calculus", and for right now, if we say that

\({ f }^{ \left( \frac { 1 }{ 2 } +\frac { 1 }{ 2 } \right) }={ f }^{ (1) }=\dfrac { d }{ dx } \)

then some fractional half differentiation operating twice on any power of \(x\) should return an ordinary differentiation of that power of \(x\). Let's suppose that fractional half differentiation of \({ x }^{ n }\) returns

\({ x }^{ n-\frac { 1 }{ 2 } }\dfrac { \Gamma (n+1) }{ \Gamma (n+\frac { 1 }{ 2 } ) } \)

then doing this twice in succession will in fact return \(n{ x }^{ n-1 }\). Because just about any mathematical expression can be expressed in power series, we can employ this definition of a fractional half differentiation and get consistent results. Getting into other fractions of differentiation and fractions of integration is another matter, and expands on this subject, but this is an already studied matter. This is an interesting subject, and it's too bad that it's not discussed more often. Who says that differentiation and integration only occur discretely? We're already familiar with fractional dimensions, why not fractional calculus? Michael Mendrin · 2 years ago

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@Michael Mendrin wwait... fractional DIMENSIONS?!



John Muradeli · 2 years ago

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@John Muradeli Yes, fractional dimensions. Get used to it. Look up fractals and fractal geometry. Michael Mendrin · 2 years ago

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@Michael Mendrin Everything I've looked up is... well... jumbo.

Do you know any web-link that substantially explains it? I don't care how rigorous it is. Oh, and if it has diagrams, that's better. Hate blocks of text.

Thx John Muradeli · 2 years ago

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@John Muradeli If you're looking for a quick read about fractal dimension and fractal geometry, have a look at this one

[Fractal Geometry] (http://mdc.nfshost.com/fractals.pdf)

but I must advise you that it's best to first independently study fractional calculus and fractional dimensions, as they've both come from very different origins. What's been happening is that as both are being independently developed, both fields are finding the other becoming useful and relevant to their own. Papers are coming out about "fractal geometry and fractional calculus", but that's not how either got started. Michael Mendrin · 2 years ago

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@Michael Mendrin Hm is Mandelbrot expansion (or zoom or function or whatever) a fractal dimensional object? Looks a lot like those John Muradeli · 2 years ago

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@Michael Mendrin Thx Michael, you the man.

So you're saying it's a new science, eh? Well, I've always wanted to develop my own calculus about dimensions... Looks like someone beat me to it >.<

But there's still hope. I'm still the one who holds the question mark... John Muradeli · 2 years ago

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@John Muradeli Well, it's not really fair or accurate to say mathematicians had "never" imagined either fractal geometry or fractional calculus "until late 20th century". In fact, mathematicians as far back in the 18th and 19th century had developed both, but it's the advent of computers that has made fractal geometry a pretty lively subject, while fractional calculus has come into the mainstream, i.e., now it has real engineering applications and consequences, again facilitated by computers. I've found another paper on fractional calculus which I think is an excellent review of the subject, its history, applications, and, most interestingly, a geometrical interpretation of what it means.

Fractional Calculus

This is typical---by the time a mathematical subject makes it to Wikipedia, it turns out mathematicians have already been studying it for over a century. This reminds me about the famed High Sierra mountaineer, Norman Clyde, who was famous for leaving evidence of first ascents, disappointing many later climbers thinking they had finally made the first ascent. When confronted with that by some such climbers, Norman said, "Oh, those really weren't first ascents. I usually find stuff left behind by local Indians before me." Michael Mendrin · 2 years ago

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@Michael Mendrin Yeah the idea of fractional calculus is so interesting, but yet I didn't encounter an idea of its applications. I mean there is no physical interpretations about it.

If there is, make it clear for us! Hasan Kassim · 2 years ago

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@Hasan Kassim You're right, the matter of physical interpretations of fractional calculus is still an open question. Nevertheless, it's an expanding new field, and it's finding its way into problems of non-linear dynamics. See this helpful article that describes it

Fractional Calculus Michael Mendrin · 2 years ago

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@Michael Mendrin Oh wait crap do I owe you the question mark?? Nah. You still have to explain \(f^{(i)}(x)\) xD I knew \(f^{(1/2)}(x)\) was feasible to at least some extent (even though I have no idea what you wrote), but not \(i\). Now that's a killer.

But yeah what about what I wrote? Is \(f^{(-1)}(x)\) considered correct to represent integration? I mean, makes a complete sense to me, but y'know... With math you never know.

Cheers, John Muradeli · 2 years ago

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@John Muradeli The example I provided for "fractional half differentiation" can be generalized to include other fractions of differentiation and integration, but I'll have to think about imaginary differentiation. That does sound like a real interesting idea! Michael Mendrin · 2 years ago

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@Michael Mendrin WHUUT? Fractional derivatives are a thing?? Hah was I so right to call you Mr. Mathopedia! Man you know everything! Can I get the Michael Mendrin app on the app store? I'd pay for that! John Muradeli · 2 years ago

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this thing is sick!!! Incredible Mind · 1 year, 8 months ago

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That would be an updated version of the Schrodinger quantum wave equation. Yeah, it's probably time. Michael Mendrin · 2 years ago

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@Michael Mendrin So this would be considered an application of fractional calculus, right? I gotta read more on the concept when I have time. John Muradeli · 2 years ago

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@John Muradeli If you'd like to be at the frontiers of mathematics with applications to engineering or physics, this would not be a bad place to look up. Michael Mendrin · 2 years ago

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I do like the term "Derintegral", it does stick to the mind, like gum sticks to the shoe. It's kind of like, "Deranged". But when generalized, the concept of differentiation and integration really is nothing more than going opposite directions on the real line. Now, let's see if it's even possible to expand that into the complex direction.

Also, "imaginary fractional calculus" is a thing too, but this is a pretty new, emerging field. Michael Mendrin · 2 years ago

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