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 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.

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

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)

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?

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 ?

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

I definitely like this video, thanks a lot

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

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

Thank you!!!

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!

Awesome!

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

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

Amazing

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

Thanks

What happened to the half constant?

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

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

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...

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

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

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

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

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?

Now to make a hyper-Taylor series with fractional derivatives

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

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

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.

Then can you use it like an operator’s?

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?

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

Ansatz

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

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

Interested

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?

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.

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

Γ関数たのちぃ

now do the half integral of sin(x)

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...

Spiel mit dem Ansatz 🤓

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 !!!

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

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

Tf, when did they invent half integrals lol

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.

What about the imaginary integral? 🙃