Laplace integral gone BANANAS 🍌
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- Опубликовано: 10 июл 2024
- Calculating integral using Laplace transforms. Evaluating integral of x^m (1-x)^n from 0 to 1 using convolution and Laplace transforms of functions, which is useful for engineering systems. I use a u sub and then apply the transform to power functions, which then turns it into a product of transforms. This is much faster and elegant than expanding and using the binomial theorem. There will be factorials involved and I also mention an extension to gamma functions. This is a must-see for calculus students, engineers, mathematicians, and differential equations lovers.
0:00 Introduction
0:50 Convolution
4:00 Laplace transform
6:07 The answer
Convolution Intuition: • Convolution Intuition
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I absolutely love calculus, but I cannot fully understand this video. Still, it was mesmerizing to watch. Your enthusiasm encourages me to keep on studying. Hopefully one day I will understand this "convoluted" concept! :)
He must have edited out the interesting part, that is the part where Laplace transforms are explained or even described
Edited out? No, it would take another video to explain the details
Not to lure you away from Peyam, but Khan Academy has a few videos explaining the workings nicely in the differential equations section. With examples. But there’s so much rich theory behind transforma. It’s cool but time consuming to dive in.
If you love calculus, you should check out differential geometry, it's basically calculus but on steroids!!!
I came to this Integral during my prep for my Stochastics exam, where it was given in a more general way using the gamma function. This identity was given, just to show, that a different function was a probability density function. It was never explained, why this identity was true. Now I know, thanks!
Interesting!
I love how much fun you have in each of these videos.
Just a quick observation: the expression at the end is the inverse of the trinomial coefficient ((M+N+1) choose (M,N,1)), so maybe there's a combinatorial proof hiding somewhere?
Right I was thinking of that too! I mean you can expand the integrand out using the binomial theorem and then integrate
That's really incredible professor, we want more tricks like this
I never understood how the beta function had a easy close form solution… thank you Dr. Peyam!
With some amazing continuous symmetry formalization that I learned from algebra 1, just by looking at this integral I can immediately tell that it is equal to itself.
Wow, amazing use of the laplace transform
I'll use this idea from now to forward. Very smart
Laplace transform!!! And you make me happy 😀
Awesome thumbnail! 👍
Excellent presentation of the topics in a simple manner. Vow !
That was a very convoluted way of declaring your membership in the Gaussian distribution. Cool!
Great exercice to start a cool day with. Thanks
"fancy channel for the fancy person, you came to the right place"
That's my only takeaway 🤣😂
And I will always fancy that.
"apply fancy L" is the absolute best way i've ever heard of a Laplace transform happening, thank you lol
I love this proof for the relationship between Beta and gamma function !!
Your videos are the highlight of my day. Your optimism and love of math is inspiring and contagious! Thank you so much!
Thanks so much!!!
Masterpiece.
Class of automation today... The professor starts explaining these things on the board! The RUclips algorithm really works in mysterious ways! Love from Italy guys🇮🇹❤️😉😎
Wow, amazing!!!
I’d be interested to see the full “genealogy” of advisors!
That was freakin' weird. I love it.
Calculus substitutions always completely baffled me (I could never remember them, for one thing), so this would have been completely beyond me even when I was taking college math courses. But it’s still amazing and beautiful! 😮👍😃
Michael didn’t mention that just about all professional mathematicians in PhD descendancy go back to Laplace and Gauss and Euler… What is more special is the enthusiasm with which the Dr explains. Now THAT is special and lovable.
Nice video. I knew that it was the beta function but l didn't know that you could use the Laplace transform to solve it. I am very pleased with the information in this video.
Genius
Nice 1 Dr Pi-M😀
Can you please do an illustrative example of the above method
Plug in any value of m and n you want, and voilà
As a French, it's nice to see "le dénouement" :D
What an amazing mathemagical “pedigree”!! I have seen Dirac and Dyson lecture or speak, and have one teacher who worked on the Manhattan Project, one who was a Doctoral student of Wigner, and a piano pedigree as well (back to the Moscow Conservatory).
Thanks so much!!
You really have LaGrange, Gauss and Laplace in your ancestral advisors? That is so cool!
Yep 😁
recognize its the variable part of beta(n+1,m+1) then adjust for constants so integrand integrates to 1 😁
Not really, then you need to prove your beta formula
Wow, Dr Peyam ,you are left-hand guy ~!😁😄
I believe that is better to treat this integral as the betta function. Which automatically gives the result in terms of factorials or gamma functions, if m and n are not integers.
But how do you know that it automatically gives the result? The beta function is just rewriting the same problem
A probabilistic approach.
Take m+n+1 uniformly random points on [0,1]. The probability of a given point to be in the m-th ascending position is given by the integral of [ C(m+n,m) x^m (1-x)^n ]. Of course this evaluates to 1/(m+n+1) because of symmetry. Reordering gives the desired result.
Moreover, setting m=n=1/2 gives a nice result namely that (1/2)!^2 = area of a circle of diameter 1.
I like that a lot!!!
You and I have something in common. We both went to a lycee francais!
I once used this probabilistic derivation to define the beta and gamma function.
@@lucatanganelli5849 Sympa!! Which one?
Interesting alternative to deriving the beta function. I'll have to remember this one.
It looks easier than the standard change of variable method using a double integral from the product of two gamma functions, but it uses lots of Laplace transform machinery. Is it really equivalent to that method?
After the drum roll… Laplace Transform!, I was totally expecting you to say, “you can hold your Lapplause until the end”
Hahaha
Just learned about the laplace function 😳
Hello, I would appreciate some contant on the weirstrass transform. Im working in a "in-network" computing environment, and the chips im on dont have division/loops/much memory so I had to cook up some specialized taylor polynomials and implement them in a special numerical type. throughout conducting reasearch for this implementation, I came across the weirstrass transform and it seems really interesting.
Thats a cool way to derive the beta function eh?
It is, I didn't even notice that at the start somehow.
I wanted the answer to the integral in the thumbnail 🤣
professor, what if we substitute x = cos^2(theta) or sin^2(theta) ??
how cool is that?.....super...super cool
There are some insane identities in Maths sometimes.
This is interesting Dr. Pi-M🤩, but please what happens if that x^m is y(x) already?
Meaning like this y(t)(x-t)^ndt inside the integral from 0 to 1
6:55 I agree, the Laplace transform is a beautiful tool, but the integration by parts recursion is actually extremely simple in this problem as the integral gets closer and closer to something easy to evaluate with each iteration of the IBP formula. Every time you do it, the uv term goes to zero and you get simply that integral[x=0,1](x^M(1-x)^N)dx=(N/M)*integral[x=0,1](x^(M+1)(1-x)^(N+1))dx. You just reiterate N times until the integral portion contains only x^(M+N+1). From there the answer pops right out. It is actually much more straightforward than lots of integrals that utilize recursions based in integration by parts.
I arrived at the answer the same way as you described. The fact the Laplace Transform turns a convolution of products into simple multiplication of individual convolutions is proven “using” integration by parts in the first place. Also, the change in variables makes the procedure rather unintuitive and/or unclear as to the motivation behind it. This particular problem doesn’t need that much advanced pieces of math.
This is the beta function. Amazing video!
can you please explain why exactly does Laplace transform work well with convolution... I haven't come to this topic yet at uni but maybe I could make sense
L{ f*g } = FG, i.e. it turns convolution integrals into multiplication. After multiplying, find the inverse laplace of the product and you get the answer to the convolution without having to integrate directly. That's essentially the end result of the video; he didn't even find an antiderivative. If you are an engineering student, you will learn that concept in your 200 level courses. Happy learning!
When is part 2 of antipythagorian theorem coming?
Sunday
@@drpeyam 👌
Beta(M+1, N+1)
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@2:10 Q: What comes after tu? A: "You're welcome" This is definitely Laplace pour math
I recognized Beta function in this integral
He showed how to express Beta function in terms of Gamma function
So, basically, I = 1/1320... It makes sense, yes. Good approach.
Can u get the gamma function from this somehow???
Edit: beta function
Consider the following monster 👹
In over 30 years of maths that is the worse tau symbol I have seen in my life. You should do some vids in solving differential equations using Laplace.
LOL
Due to 0 to 1 limit, I simply replaced x^3 (1-x)^7 to x^7 (1-x)^3. Since all terms are of the form x^n whose integral x^(n+1)/(n+1) would just be 1/(n+1)-0.
With n=3, use nth row of Pascal's triangle for (1-x)^n getting [ 1 -3 +3 -1 ] with consecutive denominators n+1 = 8,9,10,11.
Ans = 1/8 - 3 (1/9) + 3 (1/10) - 1/11 = (5+12=17)/40 - (11+3=14)/33 = (561-560) / 1320 = 1/1320 = 1/(10x11x12) = 3!/(8x9x10x11) = 7! 3! / 11!
Nice
Dr Peyam, you made a little mistake . In the final equation , the variant should be "s" (In the Laplace transform) ,not "t" . Althrough ,its final result is no problem.
what is dx extra there
?
Bananas you say?
Hahahaha
His French accent 😍
Merci 🤩
did he say that he comes out?
Yep :)
@@drpeyam good man
A fancy channel for fancy persons, like me.
Oh mais il est français ?? Didn't know until noW
Using the standard definition of Beta functions would have made this simpler...
Nope, then you would have to prove your formula
I can't hear you. There's half an integral sign in my ear 🍌👂
who would replace Eminem? @7:07
let you be Tao? nice change of variables.
a beta function.... nice approach!!
"Le dénouement", c'est Français ça :O
En effet!
do you like chicken
Yaaa very much
Yes?
@@drpeyam I luv ur videos. Amazing content. Keep motivating us towards MaThEmAtIcS
Isn't this integral is simply in form of beta function
Ok? So how do you prove the beta function formula then?
@@drpeyam mmm I'm not mathmatican so I don't know all possible ways but I know a way using double integrals and gamma function
Plus your way is can be considered as derivativation
Exactly
Boomer uncle 🤣🤣🤣
răng
😂😂😂you like hard ways. That's a beta function
It’s the Laplace Transform of t^n that makes things like √π! happening. What’s not to like?
My advisor's advisor's advisor was John Hennessy. You should be ab!e to guess the general area of my dissertation. (en.wikipedia.org/wiki/John_L._Hennessy)