I completely missed how this ended up in your vocabulary or why you're using it, but I'm loving the German lingo in there.

@drpeyam

2 жыл бұрын

I grew up in Austria 🇦🇹

@BlissToby

2 жыл бұрын

@@drpeyam ohh i didn't know that, awesome!

@Danisntsorry

2 жыл бұрын

AUSTRIA!!!!!!

@theproofessayist8441

2 жыл бұрын

@@drpeyam Hope you had plenty of nice Linzer tarts!

@drpeyam

2 жыл бұрын

@@theproofessayist8441 Of course hahahaha

As soon as those factorials came up I knew the gamma function would come in and you did not disappoint!!

I think you should have mention that derivative of sin is cos which is also shift = sin(x + pi/2), and integral of sin is -cos, which is also shift = sin(x - pi/2). So, when you do half derivative twice you get sin(x + pi/4 + pi/4) = sin(x + pi/2). Not everyone keeps in mind that cos is shift of sin.

@joansgf7515

2 жыл бұрын

I think he did mention it in one of his partial derivarives video.

@dominicellis1867

2 жыл бұрын

Therefore differential operators are just trigonometric transformations and can thusly be described using rotational matrices where theta = the angle of differentiation and A = the differential surface area.

Cool! I stumbled with fractional calculus in sophomore year by playing around with the Laplace transform.

This reminds me a little bit of Fractional Calculus (invented by some Japanese?) where you can take, say a half derivative of a function. The Gamma Function also plays a role there. I am lacking a geometric interpretation of all this, but it's cool. With fractional calculus you can also find sums of complicated infinite series but at that point, my head started to burst...

one of my favorite little functions is the shifter: f(x,n) = x^n/n! when n >= 0 otherwise 0. notice the derivative of this function in x is itself shifted in n: df(x,n)/dx = f(x,n-1). This function makes for a really clean way to define e^x = sum over all integers:n of f(x,n). And of course the derivative of this sum would be itself, since that just shifts the domain n (which ranges from -inf to +inf) by one. In this case, the half integral actually follows straight away as i^(1/2) f(x,n) = f(x,n+1/2)

@MrRyanroberson1

2 жыл бұрын

f can be redefined, as you mentioned with gamma, as: f(x,n) = x^n/gamma(n+1) when n is not a negative integer, else 0 (treating gamma(n+1) as infinity for negative integers, which is what it diverges to after all; relatedly 1/gamma(x) can be defined for all real numbers in this way)

The half integral of X becomes a major 6th in a 4:3 cross rhythm. Every single differential/complex function can be represented by a musical note with a pitch p and a rhythm r where the octave is the subscript of the pitch or the integral of the harmonic motion domain with respect to the pitch positions p_n. Therefore, p = exp(iN@pi/M) and r = exp(ln(2)*(t/(n-@) and the harmonic operator is

you've heard of the half derivative, now get ready for the HALF INTEGRAL! love this video dr peyam!

Never heard of this. Interesting. Thanks.

Descriptively a derivative is a slope, an integral is the area, so i wonder what a half integral is. Also i wonder if you could define a sqrt(2)-th integral, a i-th integral, or a (variable)y-ths intergal.

Negative half derivative. I do think it's funny how when you extend to fractional derivatives and integrals get kinda indistinguishable.

@sniperwolf50

2 жыл бұрын

Yup. That's kind of the idea. Non-integer derivatives have non-local dependency, i.e. they depend on what's happening in the general vicinity of the evaluation point and not only at its closest neighbors. This fact confers non-integer derivatives a kind of memory propriety that plain integer derivatives don't have

I definitely like this video, thanks a lot

Prof Peyam. This could be the start of whole new area of mathematics.

Thank you for your presentation about half integral and differential.😊👍 Would you mind doing half integral and differential Lambert W function antitrigonometric functions?

@theproofessayist8441

2 жыл бұрын

Oh god Lambert W function? Dr. Peyam needs to make a collab with BlackPenRedPen again!

Great explanation! I just learned of the existence of half derivatives and half integrals maybe 6 months ago. Pretty cool. What about other fractional (or even irrational) D's and I's ?

@drpeyam

2 жыл бұрын

See the playlist

@mathadventuress

2 жыл бұрын

@@drpeyam 😍😍😍😍

Thank you!!!

You should make a video on fractional PDEs dr.peyam!

Lol someone already mentioned it, but what would happen with the +C ?

@slawomirdrapinski4538

2 жыл бұрын

Reduced to atoms

Thats so cool!

Is there a geometric interpretation as derivative is the variation and integral is the area beneath curve...

What happened to the half constant?

@Icenri

2 жыл бұрын

Half integral of 0?

@Icenri

2 жыл бұрын

How about doing the half derivative of 1 = Sin^2 + Cos^2 so it would be equal to the half integral of 0?

Awesome!

Then can you use it like an operator’s?

Dr. Peyam You should look into dual numbers, and split complex numbers (also called hyperbolic numbers). j^2 =1 and з^2 = 0 and j and з are not part of the real numbers I think it is a cool concept to look into My inspiration for this is how clean Euler's formula is in that form e^jx = cosh(x) + j sinh(x) e^зx = 1 + зx = cosp(x) + з sinp(x) Where cosp(x) is the parabolic cosine and sinp(x) is the parabolic sine This is also interesting since d^2y/dx^2 = -y y = c1 cos(x) + c2 sin(x) d^2y/dx^2 = y y = c1 cosh(x) + c2 sinh(x) (This can be simplified without hyperbolic numbers) analogously d^2y/dx^2 = 0 y = c1 cosp(x) + c2 sinp(x) = c1 + c2 x Oddly enough there is a whole table on the bottom of the "length contraction" article on Wikipedia. Maybe I will go through Complex analysis and do analogous proofs with these numbers; see what they can and cannot do.

Is the half derivative of a constant = 0? I am wondering about ignoring the constant on the indefinate integral? Is there anything that could be added to the half-integral that would be lost when you take the half-derivative? Also, let's think about physical interpretation. If an integral is the area of the function under the curve, then what is the half-integral?

Amazing

It would be interesting to explore the whole _continuum_ of derivatives and antiderivatives, including all the reals between the integers. How does a function "evolve" as the degree of the derivative changes? Degree zero is always identity for all functions; f(x)=x^n for example is x^n at degree zero, and "ends up") as nx^(n-1) at degree one; f(x) = e^x doesn't change at all...

There _is_ the pesky matter of constants that Peyam doesn't talk about here. That would make an interesting video as well!

Now to make a hyper-Taylor series with fractional derivatives

How do you half integrate ln(x), arctan(x) or other fonctions ?

@nikoluk7109

2 жыл бұрын

Half-integral is a linear transformation and probably continuous(?), so one can try to expand them into a Taylor/Fourier series and change the order of series with integration.

@guitar_jero

2 жыл бұрын

You can use integral transforms such as Laplace.

Dr. Payem, can u please do a half integral of trig and exponential function and compare it with the ordinary integration

@drpeyam

2 жыл бұрын

I talk about that at the end of the video

So...the half-integral of x evaluated at π is the volume of a sphere of radius 1. Is there some intuitive reason that should be true?

@Salarr

Жыл бұрын

Kind of… There is some intuition, dimensionally, for the half-integral of the area of a circle with respect to itself, being the volume of a sphere with the same radius. If you half-integrate an area with respect to an area (linear dimension squared), the dimension of the result is an area times the linear dimension, so a volume. Try the substitution x = pi * r^2, the circle’s area. The half integral becomes 4/3 pi r^3, the sphere’s volume. The reason evaluating at pi corresponds to r = 1 is because x (= pi * r^2) = pi, so r = 1. It might be logical to use A as the variable of integration, rather than x.

So using the series expansion for the sine function and “ half integrating ” I get I½ (sin X) = 1/√π { [(4^1)/3] X^(3/2 )− [(4^2)/(3•5•7)] X^(7/2) + [(4^3)/(3•5•7•9•11)] X^(11/2) − ••• } Half integrating again gives the result for − cos X when adding the right constant of integration.

So what would be "half-constant"? When you apply half integral twice you get regular integral +C - so... this C must come from somewhere...? 🤔

@martind2520

2 жыл бұрын

This is a very important question.

@OmChougule879

2 жыл бұрын

half a constant would still be a constant

@gowanoatta8585

2 жыл бұрын

@@OmChougule879 If I½C is still a constant, then I½(I½(C)) is still a constant. But I½(I½(C)) is integral of C, which is not a constant when C≠0. There's something wrong.

@mathisnotforthefaintofheart

2 жыл бұрын

This is an intriguing question. The normal constant is explained through the Fundamental Theorem of Calculus. However with Fractional Calculus, the whole platform is different. I am not even sure as to how to interpret the FTC in this realm. That begs to question in what sense the constant even plays a role here (or not...)

This may be useful for evaluating zeta functions since they use the gamma function and are continuous.

Жыл бұрын

Well, yes, but this is (one of) the ways zeta function(s) arose. So that is done already. But surely there is more to do.

Thanks

@drpeyam

2 жыл бұрын

Thank you so much, I really appreciate it!

If I take the integral of the velocity vector of a particle I get the position vector. What does the half integral of the velocity vector tell me about a particle's motion?

@drpeyam

2 жыл бұрын

Half position 😂 Or maybe quantum mechanical position

I suppose that the exponential function is unaffected by the 1/2 (or any other fractional) operators.

@drpeyam

2 жыл бұрын

Yes

I remember seeing the 4/3sqrt(pi)^(3/2) in some diff eq problem you had the other time with like 8/3sqrt(pi)(?)

@ericthegreat7805

2 жыл бұрын

Also for that sin shift thing. Can you get the beta function in there too?!?!?

do you think this half integrals and half derivatives could be related to stochastic processes?

@drpeyam

Жыл бұрын

I wouldn’t be surprised!

Please help sir : Integrate {exp(-exp(1/x))} from 1/2 to 3/4 .

@drpeyam

2 жыл бұрын

No idea

Sqrt (pi)?! Where did sqrt(pi) come from?! Love it!

Saw it for the first time, someone just holds pen the same way as I do. (looks funny though).Btw Gamma Function is commonly used in statistics, while computing Normal Distribution and another function which we regularly use is Beta function.

Interested

Can you do a video on calculus without limits? Let dx = [[0, 1], [0, 0]] Then dx^2 is zero. I is the identity Evaluate f(I * x + dx). Then you get f'(x)dx, in other words the upper right corner of the 2x2 matrix is f'(x) Start with polynomials. You can do the derivative of exp() by using the power series expansion. So still, no limits

@drpeyam

2 жыл бұрын

Check out my video on Linear Algebra Derivative

Ansatz

I didn't know these half integral things existed. Just the other day I was wondering hey what if you put the integrals/derivatives just on a spectrum and make it a function wherein you feed a real that determines how many levels to integrate or drive. It being fractional would then just mean what the gamma is to the factorial or the fractional powers are to repeated multiplication. And here it is. Can't invent anything these days...

now do the half integral of sin(x)

@drpeyam

2 жыл бұрын

I mention this at the end

@oguzhantopaloglu9442

2 жыл бұрын

@@drpeyam yeah but you didn't calculate it?

What about a third integral or root two integral? Better, pi integral!!!

@drpeyam

2 жыл бұрын

Just let alpha be whatever you want :)

Γ関数たのちぃ

Spiel mit dem Ansatz 🤓

What is the meaning of half integration? Your nice video frustrates me, it doesn't define the notion

@drpeyam

2 жыл бұрын

I tell that at the end

@thibaut5345

2 жыл бұрын

@@drpeyam You say what it can be useful for, but I'm left wondering what it means in the first place. Maybe a separate video will define the notion of half integration 🙂

@nathanisbored

2 жыл бұрын

@@thibaut5345 he gave an explicit definition at 4:52. I believe the meaning is just to be an abstract generalization of successive integration

@thibaut5345

2 жыл бұрын

@@nathanisbored It's an application to a particular function, not a definition of what a demi integral is. He says "this is the demi integral of x" without explaining why. Who is this beast that he feeds with f(x)=x ?

@nathanisbored

2 жыл бұрын

@@thibaut5345 it looks like there are multiple further generalizations to f(x), but i suspect you would want any generalization to have the property that you can evaluate a function by its power series termwise using the formula in this video

Gosh, this mathematician Peyam ... Now, what next? Quarter integral , one third derivative, 0.45 integral ... ? May be, the symbol showing the upper portion of the usual integral sign is upper half integral, that showing lower portion the lower half integral! Question - Then, what is half integral of sin x? Replacing X with sin x, we get (sin x) to the power of 3/2 !!!

@drpeyam

2 жыл бұрын

It’s actually sin(x - pi/4), as you can see in my half derivative of sin video

@TheAzwxecrv

2 жыл бұрын

@@drpeyam yes, i realized that after. It is the half integral (sin x) d(sin x) that will give (sin x) to the 3/2.

Tf, when did they invent half integrals lol

@mathisnotforthefaintofheart

2 жыл бұрын

One can imagine to invent all sorts of wonderful things if you stay away from social media😆

You didn’t even define what half integral represents. You just postulated the integral number integration formula. You can’t do that while making sense.

@drpeyam

2 жыл бұрын

It’s a nice Ansatz

What about the imaginary integral? 🙃

@drpeyam

2 жыл бұрын

Alpha = i