Feynman's technique is the greatest integration method of all time
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- Опубліковано 20 бер 2023
- Another beast of an integral laid to rest by the sword of Feynman!!!
The solution development is absolutely gorgeous and the result is surprisingly satisfying.
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Young mathematically talented kids these days are so lucky to have the internet as a resource to keep them stimulated. This kind of video is exactly what I needed as a young teenager.
As a teenage self-proclaimed math goblin / Feynman acolyte, I concur.
Most of the current IMO participants also watch a lot of math videos.
As fourth of Europe at the IMO last year, I am surprised how much there is to learn on the internet.
I feel so jealous of them 😁
I wish I had access to resources of this king when I was young. I grew in a village with no books and libraries. I barely had a blackboard with some pieces of chalk and a kerosene lamp that hurt my eyes at night during homework. But somehow I took pleasure in math.
@@slavinojunepri7648 where did you grow up?
The more I watch feynmann integration technique videos, the more powerful I become.
Same!!
@@azizbekurmonov6278 азизбек.не русскоговорящий ты случайно?
@@Dagestanidude Da ya panimayu
Lol
Xp farming on this video
It's been 50 years since I've solved a complex integral. This guy moves too fast for me! I'm reminded of my old teacher, and later friend, Wolfram Stadler. Rest in Peace, Wolf.
Ditto. Learned how, then never had to use them again. Today, fugetaboutit!
sir, may I ask what you studied and what you did in your professional career? I'm planning to get back to grad school for math and computing
@@LetsbeHonest97-- If you're asking me, I earned an undergrad in EE in 1980 and a master's in CS in 1984. Go and do it as soon as you can -- school gets more difficult as you age.
@@kwgm8578 absolutely ... Will do asap
@@LetsbeHonest97 Good luck to you!
So to sum it up and generalize:
Craftily plug in a parameter a so the derivative of the integrand with respect to a is simpler, now you have I(a) and you're looking for I = I(a0)
Derive the integral with respect to the parameter making sure swapping places between the integral and the derivative is allowed (check convergence)
Make your way towards an explicit expression for I'(a)
Integrate I'(a) yielding an extra constant in the I(a) expression
Determine the constant by plugging in I(a) a nice value for a making it trivial to compute
Replace a by a0 and voilà, I(a0) à-la-Feynman, serve hot with a light Chianti.
No wonder they use a math sign language. What a ride!
Hero
My summary:
Find someone better at math than me and ask them for help. Maybe I'll find this guy's email somewhere...
We makin it outa Cornell wit dis one😎
Excellent work, a good way to check the answer is by plotting the function (e^-x^2)*sin(x^2)/x^2 and estimating the area from 0 to infinity under the curve. The function is > 0 from x=(0 to 1.722), and the function is almost zero for x=(1.722 to 2.35) and then zero for all values of x>2.35. You can approximate the area under the curve as a right tringles with sides of 1 and 1.722. The area for that right triangle is (1x 1.722)/2=0.861. The exact answer per the video is 0.806626.
Love how you talk about mathematics with passion while solving :)
My favorite aspect of Feynman is that, while he was certainly a genius, he has a big dose of ordinary guy that we can relate to. I'm not in his league by a long shot, but I bet it would have been a blast to hang out with him.
With respect, what are you talking about lol? 😂 Feynman's brilliance was only matched by his ego and capability to be a complete asshole. His lecture series are engaging and make him out to be what youre trying to portray, but the reality of his personality was quite a bit more grim in both nature and circumstance of his life. He was a good teacher; as that tied into his work, but no you really wouldnt want to be "buds" with him and he most certainly is not a strong candidate for representing the "every man". Sorry to burst your bubble; but best to keep his legacy wrapped in his brilliance and contributions to science as a whole, not his personality.
Surely you're joking, Mr Feynman... ;)
JgHaverty, spoken like a true ignoramus.
@@TheSireverard, and also "What Do You Care What Other People Think?"
@jamesedwards6173 what the hell are you talking about? Hahaha
epic , thank you for making this technique so clear
Wow yes this is so intuitive and elegant and beautiful and I totally followed you the whole way along
Thanks so much 😊
Noticing that d/dx(-exp(-x^2)/x) = 2exp(-x^2) + exp(-x^2)/x^2, I went for an integration by parts, which also works nicely, but is less elegant I admit.
I found amusing that in that case, the result appears in the form of sqrt(Pi/sqrt(2))(cos(Pi/8) - sin(Pi/8)). After multiple careful checks for mistakes, I eventually realized it is actually the same result as in the video!
In the video is =d/da[sin((ax²) dx =f of d/da
X² ½-a
The -exp =to its integral, but its sin8 and exp
This was amazing, really gotta use it instead of by parts. Thanks a lot !
This is AMAZING!! Thank you for your great video. I think I lack some basic techniques regarding imaginary number but except that everything was super clear and easy.
This is advanced
Been waiting for an explanation of my favorite’s, Feynman, noble prize topic.
Absolutely beautiful. Thank you for sharing!!
Beautifully done video!
Cool video. :D
Another way I think you could do is using my #1 favorite method, ha ha. Once you've differentiated and the integrand is in the cosine form, use Euler's definition to re-write cos. Then you have a sum of integrals of exponentials. Then the trick is, make a u subsitution for the argument of the exponential, that puts the integrals into the form of a Euler's integral definition of gamma. The power of u allows you to determine each z.
Amazing! I solved this by defining an I(a,b) equal to the integral with a parameter inside the e and the cos. Then differentiating partially and adding to get a first order PDE. Then conjugating and using partial integration to get the required result!
Your method is much slicker, as you just took the real part rather than dealing with the whole complex function!… 😂
It's crazy
As someone who failed their A level maths almost forth years ago, I found this video utterly fascinating and understood (or rather, could follow) practically none of it . . . .
Did it (after seeing video) with the a on the exponential term.....follows pretty much the same route except using the Im operator as sin(x^2) is a constant. Other than proving Im(sin(x^2) = 0) over the range, pleasingly we get the same answer.
very perfect, I tried to do it myself and needed the video again and again. But now I got it all. See research gate if you are missing 2 or 5 steps in between.
We used to study similar integrals using the residue theory in the complex field and the polar coordinates.
Nice integral! I wonder if it's solvable putting the a parameter into the exponential instead? Seems like you should end up at the same place. To solve the constant of integration you would need to let a tend to Infinity instead of setting it to zero, and the rest should be the same.
I'd imagine you'd get issues with the fact you'd still have the sin and therfore a complex exponential which makes things more complicated
@@patrick-kees8962 I believe it would still work if you consider the Imaginary part of the integral instead of the Real part
This may be one of Feynman’s integration techniques (he has several and needed them to perform integrations necessary to compute Feynman diagram calculations) but it isnt the one he was most famous for…. Integrating by analogy with finite summations and vice versa. This particular technique, or parts of it (particularly integration by differentiating under the integral sign) is discussed in Engineering Mathematics Advanced texts such as Sokolnikoff & Sokolnikoff . This particular calculation is a bit more involved as complex variables are introduced
You're doing really good content. Please, moreeeeee Feynman Integrals!!
What a beautiful integral! You might also be able to solve this same integral using residues/contour integration.
Been listening to the Feynman audiobook ("Surely...") and Feynman was a PLAYA wowwww. Dude got around! And then he talks about this, so I had to look it up. I've only taken Calc 1, so this is way beyond me but fun to watch. I'll have to watch more videos to understand it better.
Wow. This technique is amazing. Maybe not even among the top 10 achievements of Richard Feynman but still fantastic!
This makes me want to learn complex analysis. Great video considering I still understood most of it
Very cool! Thanks for sharing.
Very nice presentation.
Thank you Sir for your best explanation and working out of the problem🥰😍🤩
Thank you for the nice comment
very nice effort. good luck
Can't wait to learn all this it seems interesting enough 🙂
Amazing content!
Great video. Thank you
Brilliant! Thank you.
technically you also have to ensure that the differentiation and integration are interchangeable (which is not true in general for integrable functions) which can be quite tedious, especially when working with improper integrals
He covered that in the video, albeit somewhat handwavingly.
@@thomasdalton1508 Yes. The handwaving ignored the potential problem at the left-hand side, where x=0 and x^2 is in the denominator. It's fine, but should be addressed.
@@egdunne It doesn't need to converge at x=0 does it? The integral is from 0 to infinity, so it needs to converge on the *open* interval (0, infinity). The boundary points don't matter.
Mathematicians will worry about that, physicists not so much.
@@evertvanderhik5774 Physicists might not worry about proving rigorously that it converges appropriately, but they need to worry about whether it does or not otherwise they'll get the wrong answer. You can determine that using rules of thumb rather than a rigorous analysis, but you have to do it.
I came up with this myself in college. I hadn't known until now that this Feynman guy stole it.
😂😂😂
I completely believe you
I'm one of the very unlucky ones who are incapable of math beyond basic algebra but am fascinated by it. I watched the entire video despite understanding nothing.
I'm not sure if this is just an elaborate form of self-harm...
Absolutely. I feel exactly the same!
Why did we stop? application of a formula for the cosine of double angle shows that sin(pi/8) equals sqrt(2-sqrt(2))/2 ... which allows us to simplify the entire answer to sqrt( pi (sqrt(2) - 1) / 2) ; that final formula does not use any trig functions (sin,cos,etc). Just a thought :)
Once upon a time I would have been able to reproduce this. Now I am just watching and thinking wow.
Just infinitely beautiful!
SUIIIIIIIIIIIIIIII
I just want to know which drawing tablet do you use for mathematics and which app (on Android Tablet I suppose) ?. Thank you very much. And great content!
I love this video!!
Almost everything is cool, except for one. Complex numbers have two square roots. It would be nice to mention this and show that it does not affect the result.
It is a minor omission, but you are right
@@svetlanapodkolzina1081 It's not a minor omition, we don't have logarithm complex function because of monodromy. It's impossible to define square root on all of C.
Beautiful!
I have a great integral as an idea for a video
The integral from 0 to ∞ of e^(A(x^B))
Where A and B are any complex numbers except the values of divergencey and to find what are they
You are mad man indeed ... You mad a great Difference. So clever...❤❤❤❤❤
Beautiful solution
This is sheer brilliance. I found something with a similar message, and it was beyond words. "The Art of Meaningful Relationships in the 21st Century" by Leo Flint
I understood it but it still made my head spin!
nice demonstration 👍
Why stop there? If you evaluate sin(pi/8) further, you can write the result as sqrt(pi*(sqrt(2)-1)/2), which I think is quite nice.
I liked the sine term at the end but yeah the radicals are quite nice too
Radical!
Me too.
Thanks you , greeting from Argentina.
Please tell me why we take just real part in 3:43. I see that we need just cos but I do not undersfand how can we ingore sin part of Eular formula.
Very awesome technique, I love it - great👌
Calling it Feynman's technique makes it appear as though it took centuries to develop it, when in reality this is also known as Leibniz's rule after one of the creators of integral calculus, so it was actually known pretty much since integration became a thing.
Hey, just to add to your knowledge the lebinitz rule basically deals with differentiating a function under integration, whereas Feynman's techinque is a way to find definite integrals of non integrable functions by introduction of a parameter while 'using' the lebinitz rule as a smart tool and hence " lebinitz rule is different from Feynman's techinque, one helps the other."
Nah Leibnitz rule is different.
Great video, primers are so much better than triggers
At 5:00. This integral can be determined easily by switching to a 2D integral in polar coordinates. No need to use formulas from books.
Around minute 10, you can just use the fact that 1-i has angle -π/4 so the square root has half that, and multiplying by i rotates it by π/2 meaning that the new real part(cosine) is the old imaginary part(sine). Just seems slightly easier and more intuitive than the algebraic argument.
Instead of -pi/4 i used 2pi-pi4=7pi/4 which is the same but got different answer. 😢
Nice video. What application and writing device(pen) are you using to write so nicely math?
I would like to know also.
I feel that im evolving after watching this!!
can u make a video about the feynman technique itself ?
My AP calculus BC brain has expanded… glad i’m pursuing a stem major 😃
Eh this is pretty entry level stuff on tbe grand scheme of things. If you really want to "expand your brain", go noodle around feynman diagrams; with regards to path integrals and quantization 😅. If you REALLLY wanna see where this rabbit hole can go, then go over neutron transport while youre there 😂
Recommend calming the hubris of your AP calculus class. The reality is if youre pursuing a degree in engineering, physics, or whatnot; your best interest is actually not using AP credits for anything other than humanities. Encumbent on what programs you narrow down and get accepted to of course [if your program only requires calc 1, then yes of course use your ap credit in that capacity]. Its a good path to be on; just take it in stride. That said, AP credits are kind of useless beyond gpa padding and i dont understand why highschools put so much weight on them in the first place..
This technique is elegant but can it be solved using complex integration involving cauchy residue theorem?
And a lot more easily
Nice video!
Fascinating technique, are there applications of this integral?
Can feynman's methhod be used for all integrals? If not, what are the restrictions please?
The square root in complex numbers has two solutions. You also have e^7pi/8 as solution
Honestly, using Re on euler's theorem that way is more impressive than feynman's technique, imo.
That's precisely the sort of chicanery that i started to love these subjects for!
edit: first time I saw that integral was statistical mechanics and the professor just gave the formula without proof or derivation. In numerical methods we got to see montecarlo integration, and that's probably my favourite integration method. Didn't see any of this in complex variables, which I went on to fail.
It would be easy for me to love mathematics if my teachers were like you!
Why choose to throw alpha into the sine function as opposed to the x^{2} in the denominator or the exponent exp{-x^{2}} in the numerator?
shouldn't the -i be in the numerator after you solved int I'(a) da by substitution, hence providing the neg solution to that integral? sry if i am wrong, it has been some time...
Which app you use for writing please tell me
beautiful
What is the ñame of the program tiene récord the video From your mobile ?
Wonderful!
What are the constrains on fx when using Feynmans method - you mentioned convergence and considered if the function was increasing or not
Also does it only work with fx with bounds 0-infinity?
I'd like to ask what's the device you record on? 👀
How u define time?
amazing
Where did the pi under the first radical come from in the last line? Shouldn't it just be root 2 of root 2 multiplied by sin pi/8?
Can this be done using laplace transform ?
I love Feynman Integration! Why isn't this taught in undergraduate?
because it's hard to predict what the parameter a is, and where you should put it? That's the Satan's level mate !
@@yassinetiaret505 so you are saying it's too hard to be taught for college students 🙃
it is in my program
It is in upper level Physics classes
I think some of the math involved in this problem isn’t undergraduate level math, unless you’re a math major.
For example, I don’t know much about a lot of the things he did with the imaginary numbers except from an identity we used in differential equations.
I like the pace, you don't go at a snail's pace like some others. Great job!
I'm never comfortable with just discarding the "i*sinx" part, especially when the cosine can be defined as (e^(ix) + e^(-ix))/2, no discarding of terms required. But the math would proceed much the same either way.
Discarding makes it simpler
Integral calculus is already difficult do not invent new obstacles for yourself :)
with complex numbers this is totally okay because they have a real part and an imaginary part. If we're looking for the real part then there is a 0% probability to make any mistakes by leaving out the complex part in instances like this. You can obviously still make calculus errors etc. but that wasn't the issue here.
@@CeRz I guess I'm good with dropping the imaginary part at the very last step, but not before that.
@@kingbeauregard and that is totally fine. However, if you ever change your mind for optimal efficiency you're still aware that it is possible to execute it like this aswell. To each their own. Good day.
that passion about maths =) I could feel it
@maths_505 what is the name of the software you are using to write the stuff down?
It's my galaxy note phone...
Nothing else
@@maths_505 thanks
10:24, I think we have two cases: -π/8 or 7π/8. But for case 7π/8, we can find that the final result of the intergration is negative which is impossible.
Why impossible? The function is sometimes positive and sometimes negative
to the guy above me, no, the integer is a positive series, and can never be negative because of 0 to the positive infinity.
3:49 Hi! What is that method called, it seems very interesting.
Why does it work with the other exponential in the integrand top?
Oh that's just Euler's formula
@@maths_505 Oh no what I meant is: why can you take the real part outside of the integral? Why does it still work even though there are other terms inside of the integral?
It works because e^-x² always gives only real values in the integration interval, right?
@@giuseppenonna2148 the real part operator and the integral operator are interchangeable
@@maths_505 I see, thanks :)
Awesome!
sin(pi/8) is easy to calculate:
sqrt((sqrt(2)-1)/sqrt(2))/sqrt(2).
Hence, we can simplify the result:
I = sqrt(pi/2) * sqrt(sqrt(2)-1)
How do we know that differentiating with respect to "a" wont change the value of the integral?
Can you please tell me the name of the tool you use to write like this? Also tell me the name of the software please
Samsung notes
Intégration by paramètre it is really powerful method.
But it isn't Feynman's method.
Wonderful solution! Can someone explain me why is a=1 our target case? I feel like I missed something. I understood that a=0 is essential to find the constant of integration but couldn't understand why a=1 is our target case? Thank you for your explanation! :)
putting a=1 gives us the original integral we were supposed to find. sin(ax^2) becomes sin(x^2) again
@@kushal_2oo4 Thanks a lot man! The problem discussed so much about having'a' in it that I totally forgot that the original problem didn't have 'a' in it. My bad! Thanks a lot :)
@@yashsethi1110 any question is always a good question. Now that you've come to a realization of your original problem, the solution is now engrained in the structure of your brain and you will most likely never forget it. This is the invaluable part of realizing things you never understood before.
@@CeRz Thank you for your kind words. I really appreciate them. I definitely won't forget the solution anytime soon :)
The reason why you can introduce the derivative into the integral is because the integration limits aren’t functions (Leibniz theorem)
Yess....precisely
Not exactly
You have to make sure the integral function converges. For that you can apply tests like Dirichlet's test or just look at a graph.
Nice video and cool trick. I've never seen integrals written like a rho though. ;)
Good job though.
Can i solve any integral by this method or just the ones with limits as infinity especelly asking as a highschool student
I'm gonna upload one today that has finite limits.