Gaussian integral of this kind in its "complicated version" (with the "a" power already plugged-in) appears frequently with the following usage: 1. in quantum mechanics. to find the probability density of the ground state of the harmonic oscillator; 2. in the path integral formulation, to find the propagator of the harmonic oscillator; 3. and in statistical mechanics to find its partition function (by the way quantum mechanics is statistical mechanics in its essence); The first two are also used in Electronics & Telecommunications Engineering for Oscillators & Filters, and then they are usually followed by: 4. Hermite Functions that contains the very similar derivatives you used to find the solution; 5. Laguerre polynomials; I hope one day you could give us some of your videos on these last two. Excellent video, thank you so much

@Ryokusei2

Жыл бұрын

Regarding statistics, it's just the kernel of a Normal distribution with parameters (0,1/2a). And knowing this it's basically done, because the integral of any distribution is 1 (by definition)

@alejrandom6592

Жыл бұрын

Thx ❤

"Thanks for watching and let's solve a difficult integral" has to be, BY VERY FAR, the best yt intro of all time.

That was beautiful, and most elegantly explained.

I just did that, since the integrand is even, we can write it as 2× the integral from 0 to ∞. From there, u=x² and du=2xdx gives the integral from 0 to ∞ of u^(n-½) e^(-u) du which is just (n-½)!, so for n=4 that's (3.5)!. You can get the same answer form as yours using n!=n(n-1)! and that (1/2)!=(√π)/2

This is crazy, enough to make you wonder why you couldn't see this quite elegant solution. Math isn't just a science, it's an art.

Nice video! I think you can also show the integral is equal to gamma(n+1/2) by reducing the integral to the gamma function.

Thanks

@drpeyam

Жыл бұрын

Thanks for the super thanks 😊

Very elegantly explained

That was a great technique. Keep on posting

Just WOW!

Great! Très élégant et précis!

Brilliant!

Superb!

Wow Dr Peyam- Very beautifully explained. Guass would have hugged you🥰

I love when mathematicians say "and this probably has some applications in quantum mechanics"

Mind blowing, as usual with Dr Peyam: complicated things made so simple that you litterally forget about the inherent complexity. I am sure that the greatest mathematician, i.e. Euler, would have appreciated this way of thinking because in some ways he shared it too. It is that kind of spirit that allowed him to establsih a totally unexpected relation between the infinite sum of integer inverses to some complex power and an infinite product whose terms are only built on inversed primes to the same power, to me one of the most astonishing results in number theory. Anyway thank you Dr Peyam for your didactic skills, in effect you did reverse the initial claim: pourquoi faire simple quand on peut faire compliqué.

Votre français est parfait, sans accent, digne d'un agent secret

@pzorba7512

Жыл бұрын

As tu appris le français avec une jeune française en séchant les cours de maths quand tu étais à la fac? Si c'est le cas, c'est la meilleure école et pas besoin de réformes des programmes ni de ministre de l'éducation.

As Dilbert pointed out (seems to be Dilbert?) ... anyway the particular integral, which was the topic of this video, and as Dr. P. mentioned was from "...quantum mechanics homework is for getting (or can be used for getting) the probability density for the 8th state of a quantum harmonic oscillator. To explain....the wavefuntions (got by solving the Schrodinger equation for a quantum harmonic oscillator) contain Hermite polynomials of order n (At least when you solve the Schrodinger using series.) Then to 'flesh' out the probability density... you've got the Guassian in there to satisfy the boundary conditions at infinity. And that Gaussian is multiplied by a Hermite polynomial of order n... where n is the quantum number. So n = 8 is for the eighth quantum state. (Which has n+1 = 9 "wiggles.") Oddly enough, for n is odd, in the beast, the integral is zero, for mathy type reasons.... but physically because the "wiggles" on the left infinity, cancels the right infinity.

@willthecat3861

Жыл бұрын

People have be thinking about 'things on springs' for centuries... perhaps as told by history of physics lore, beginning with Hooke. Yet it's remarkable to me that with a few (albeit and seemingly very non-intuitive assumptions) about how a 'thing on a spring' differs from a "quantum thing on a quantum spring" a good deal of quantum mechanics pops out of the math. A lot of the math has been around since the early 1800s... possibly longer. Anyway, I have enough brain burning with quantum mechanics without trying to understand "history of math" too. I'm just a the average cat! Or is that the average "Joe?'

Amazing!

Thanks You Dr.PEYAM,I WISH YOU GOOD NEW YEAR,THANKS FOR ALL YOUR TEACHINGS!!

@drpeyam

Жыл бұрын

Thank you!!!

Brilliantly explained!

@drpeyam

Жыл бұрын

Thank you!!!

Trop bien. C'est une intégrale sympa !

when someone ask me why are you studying physics, i'll show him this video so satisfying way to solve this kind of integrals, that leads you to geometry (where pi shows up from the gaussian integral is just an eyegasm). Thank you Dr π^m

there was a lot of cool tricks in this integral but my favourite one was the way you wrote odd number factorials. I remember we did it with double factorial (!!) symbol to to denote 'special' factorials

Feynman is truly a genius!

Very nice. Another approach is to use the new variable y= x^2 (integrate now from 0 to inf. ) . Then , one is directly led to Gamma[n+1/2] . But your calculation shows what this value is .

It is also used to calculate the ratio between oxilated quantum titrate and hyper-Byron crystals to make portal gun fluid. This obviously helps you transport things and people across dimensions as seen in Rick and Morty

Very cool

Very wonderful result. C'est aussi beau quand c'est compliqué

As statistician, whenever I saw "integral of e^-ax2", my sensors are automatically activated to integrate it using the Normal/Gaussian distribution hahaha

Wow genial

Your French is really good, like your videos !!

Grand respect Dr Peyam J ai toujours apprecie votre travail , mais cette fois ci , c'est vraiment une bombe concernant le Quantique ( Maths Appliquees). Merci Dr Peyam

@drpeyam

Жыл бұрын

Merci beaucoup 😁

@cherkicherki2286

Жыл бұрын

@@drpeyam Bienvenue

integrate functions on infinite dimensional spacs next! functional integral!

3:08 I call it "proof by optimism". Just assume you can do something, and if you don't run into any problems, it must have been something you could do in the first place, and that completes the proof!

One day, I hope maths content creator will prove dominated convergence properly or any type of hypothesis required before using these swapping theorems :( It’s part of the fun, and it makes the maths more complete!

@drpeyam

Жыл бұрын

Too long though

Beautiful integration

Liked it very much .It's a pity that one life time is not enough for maths let alone maths+physics.

I used moment generating functions, differential equation and some induction to solve ot and believe me it wasn't that complicated

If this is found in QM, you can probably find the context in Arfken.

You are so neat. You make it very elegant. My two cents: whenever I see an integral with an x term mutiplying by some e^x term I always remember the trick with gamma distributions used a lot in statistics courses. Anyway, it's been a long time I don't use that trick so I'm not sure whether it would work.

Congratulations!! Can you calculate the integral of x^44*e^(-(x)^30) from zero to infinity😉? I know the answer to this question

Firstly Integral of even function Secondly substitution u=x^2 answer Γ(4.5)

@drpeyam

Жыл бұрын

But that’s just circular reasoning

@holyshit922

Жыл бұрын

@@drpeyam I dont see circular reasoning here. By the way de L'Hospital is more circular than that and you overuse it

@drpeyam

Жыл бұрын

No

Agha Payam, this one was really nice!

The way to solve this problem is to make it more difficult! 😂

Right in time I was actually trying to figure out this integral on my own but here comes Super Peyam to the rescue :sob:

I think that it would be more beutiful if you used double factorial

Can you make that more specier with e^-x^2n 😉🤭

as N tends to infinity, does the integral vales converge to any finite value?

Great! What about if N= odd?

@joaovictormacedosales2520

Жыл бұрын

If you mean odd exponents for x (2N+1 instead of 2N), then the integral vanishes by symmetry: exp(-ax²) is an even function, and x to an odd power is odd, so the integrand is an odd function of x.

Or You could use double factorial :) N!!

Ey Doc! Why the name of de Integer is "Quatum Integral"?

Körper! 🤩

Hi Dr. Peyam! Kind of unrelated, but I've been watching many of your videos and wondered, what do you know blackpenredpen of? What is your relation? He is in many of your videos either behind the camera or appearing.

@theproofessayist8441

Жыл бұрын

You can check out Q&A but in short they both went to UCLA Berkley and ended up as classmates for math - now longtime friends.

An interesting challenge: 1) Study the convergence of int(from 0 to inf) (1 - cos x)/(x^[5/2] ) dx 2) Prove that int(from 1 to inf) (x^[1/x] - 1)*(sin x)^3 dx converges Dare you solve it?? :D

@drpeyam

Жыл бұрын

It sounds more like a homework problem than a challenge 😅

Again this sentence???😁😁😁 in Portuguese: "Por que simplificar se pode complicar?"

I thought you were going to do something with (1/2)!, which is √π/2.

@drpeyam

Жыл бұрын

I’m fancier than that 😜

(1/2)(3/2)...((2n-1)/2) is equal to gamma((2n+1)/2)/gamma(1/2), so you have this times √π. But √π is precisely gamma(1/2). Your result is easierly written as gamma((2n+1)/2). Even better, Actually. You simply can substitute x²=y in the original integral, writing it 2 ∫{from 0 to infinity}. Then it reduces directly to the Gamma distribution integral 2(1/2)∫y^n-1/2 e^-y dy, yielding gamma(n+1/2) i.e. abovesaid result. No need for those derivatives and factorials. But hey! Pourquoi faire simple quand on peut faire compliqué ?😊

@drpeyam

Жыл бұрын

Kind of circular reasoning

Waw

I dont understand how when the denominator had 2*4*...*(2N) you could pull a whole 2^N out of that? Didn't you mean to just pull out a single 2 which would leave 2 * N! ? Therefore the final denominator should have been 2^(N+1) * (N!)

@drpeyam

Жыл бұрын

2 x 1 x 2 x 2 x 2 x 3 x … x 2 x n

@jaredbaum

Жыл бұрын

@@drpeyam ohhh i see, factoring out from a product is def different than a sum. Thank you

As Alex pointed out.. with "... Guassian integral of this kind..." but wait... I thought his name was Dilbert... but, lol what do I know... I'm a cat! Anyway in the catverse we call these integrals (the beast at time = 0.12) a "similar Gaussian Integral." Mostly for reasons... like the prof made us... but, also historical ones. Namely the idea that a Guassian integral has only a particular exponential function in the integrand. (Too lazy to type it out.) Anyway, there's no factor as function of x.

@willthecat3861

Жыл бұрын

Or just an integral which is "similar" to a Gaussian integral, where similar is not the same.

why make it easier when it can be more complicated...in other words, just good ole adhd...

Differential operator of a variable is very rapid , I stupidly would have integrated by parts !! am always offended at Bell curve being called Gaussian it's Laplacian who calculated it's normed area square root of pi in 1780s before Gauss born by polar coordinates ( which he invented to separate variables in the Laplacian of physics ) indeed derivation of Bell curve from binomial called the De Moivre-Laplace Thereom see Proofwiki . Gauss a notorious thief Gaussian credit, Lobachevskys non-Euclidean geometry, would have stolen Galois groups had Galois lived which he said was gonna send to Gauss . However Gauss original outstanding in the hypergeometric differential equation generating All Special Functions.... log exp binomial Bell curve etc. Also modular clock forms for edge and compass construction of new polygons eg 17 sided

@willthecat3861

Жыл бұрын

Gauss was a lot of things... so it appears historically. One of which was a genius. We could do without thieves... that is more or less certain... yet; could we have done without Guass? Yes another wacky counterfactual to consider... if one dares!

You pronounce French better than English? What is your mother tongue?

@drpeyam

Жыл бұрын

German 😂

First