Half integral of x
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- Опубликовано: 30 июл 2024
- Defining the half integral, which is halfway between a function and its integral. Done by analogy with half derivatives, by finding formulas for the derivatives of power functions. It uses the factorial which is then generalized to the gamma function with Gaussian integrals. I also mention trig functions and applications to broken processes. This is a delight for any calculus student or anyone who is interested in fractional derivatives.
0:00 Introduction
0:30 Analogy
2:58 Gamma Function
5:00 Half integral of x
9:37 Properties and Applications
Half Derivative: • Half Derivative of x
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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.
I grew up in Austria 🇦🇹
@@drpeyam ohh i didn't know that, awesome!
AUSTRIA!!!!!!
@@drpeyam Hope you had plenty of nice Linzer tarts!
@@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.
I think he did mention it in one of his partial derivarives video.
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)
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
Never heard of this. Interesting. Thanks.
Thank you for your presentation about half integral and differential.😊👍
Would you mind doing half integral and differential Lambert W function antitrigonometric functions?
Oh god Lambert W function? Dr. Peyam needs to make a collab with BlackPenRedPen again!
you've heard of the half derivative, now get ready for the HALF INTEGRAL! love this video dr peyam!
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 ?
See the playlist
@@drpeyam 😍😍😍😍
I definitely like this video, thanks a lot
Thank you!!!
Negative half derivative. I do think it's funny how when you extend to fractional derivatives and integrals get kinda indistinguishable.
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
Prof Peyam. This could be the start of whole new area of mathematics.
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.
Thats so cool!
You should make a video on fractional PDEs dr.peyam!
Awesome!
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.
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...
Amazing
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?
Is there a geometric interpretation as derivative is the variation and integral is the area beneath curve...
Dr. Payem, can u please do a half integral of trig and exponential function and compare it with the ordinary integration
I talk about that at the end of the video
There _is_ the pesky matter of constants that Peyam doesn't talk about here. That would make an interesting video as well!
This may be useful for evaluating zeta functions since they use the gamma function and are continuous.
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.
Lol someone already mentioned it, but what would happen with the +C ?
Reduced to atoms
Thanks
Thank you so much, I really appreciate it!
Then can you use it like an operator’s?
What happened to the half constant?
Half integral of 0?
How about doing the half derivative of 1 = Sin^2 + Cos^2 so it would be equal to the half integral of 0?
I remember seeing the 4/3sqrt(pi)^(3/2) in some diff eq problem you had the other time with like 8/3sqrt(pi)(?)
Also for that sin shift thing. Can you get the beta function in there too?!?!?
Now to make a hyper-Taylor series with fractional derivatives
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.
I suppose that the exponential function is unaffected by the 1/2 (or any other fractional) operators.
Yes
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
Check out my video on Linear Algebra Derivative
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?
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.
Sqrt (pi)?! Where did sqrt(pi) come from?!
Love it!
Interested
So what would be "half-constant"?
When you apply half integral twice you get regular integral +C - so... this C must come from somewhere...? 🤔
This is a very important question.
half a constant would still be a constant
@@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.
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...)
How do you half integrate ln(x), arctan(x) or other fonctions ?
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.
You can use integral transforms such as Laplace.
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.
Ansatz
do you think this half integrals and half derivatives could be related to stochastic processes?
I wouldn’t be surprised!
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?
Half position 😂 Or maybe quantum mechanical position
Please help sir :
Integrate {exp(-exp(1/x))} from 1/2 to 3/4 .
No idea
Γ関数たのちぃ
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)
I mention this at the end
@@drpeyam yeah but you didn't calculate it?
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 !!!
It’s actually sin(x - pi/4), as you can see in my half derivative of sin video
@@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.
Spiel mit dem Ansatz 🤓
What about a third integral or root two integral? Better, pi integral!!!
Just let alpha be whatever you want :)
What is the meaning of half integration? Your nice video frustrates me, it doesn't define the notion
I tell that at the end
@@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 🙂
@@thibaut5345 he gave an explicit definition at 4:52. I believe the meaning is just to be an abstract generalization of successive integration
@@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 ?
@@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
Tf, when did they invent half integrals lol
One can imagine to invent all sorts of wonderful things if you stay away from social media😆
What about the imaginary integral? 🙃
Alpha = i
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.
It’s a nice Ansatz