Could this integral be solved more simply by contour integration? I was thinking that you can break the (sinx)^2 into e^(2ix) + e^(-2ix) - 2 /(-4). Close in the upper-halfplane for the first piece, bottom half-plane for the second piece, and either direction for the third piece. Observe that the integral along the real number line from -infinty to 0 is the same as from 0 to +infinity (as the function is even)
I agree. I solved it much more quickly using a standard method. By the way, the solution given on the vídeo is just one of the many solutions possible, wich appear naturally in the standard method.
Mark Saving Papa flammy likes to do it the cool way, without complex analysis. I bet its wayyyyyy easier with complex analysis (which I cant do yet so im forced to do it the cool way). Do you have any recommendations for any online sources or books I could get to learn complex analysis? I havent been able to find a good one.
@@categorille8330 Yes I got a book on complex analysis a month ago. Its called complex variables and applications by Ruel V. Churchill. I havent gotten that far into it but its pretty good for an introductory book in complex analysis from what I've seen so far.
ok so I wanted to refresh myself on both Laplace transforms and Feynman's method and was about to look both up separately but I thought to myself I will watch one more meme integral video... yet here i find myself. thank you
This is what I mean when I tell my friends that mathematics is beautiful. Results like these, those that just pop out of the blue and completely makes you unable to talk, those are the best moments you'll ever have. Beautifully done and indeed, a very beautiful integral which definitely beautified my Saturday evening =)
There needs to be more people like you; salivating over Putnam problems alone is a sad life. I wish for a world where all RUclips thumbnails are as glorious as the one for this video
I absolutely love this video. This solution was absolutely ingenious , and the way everything simplified so nicely in the end weirdly pleases me. You actually did make my weekend, and as a result I am now a patreon. Keep up the incredible work!
I can't understand why you're not a +1M subscribers channel, your content is so great, so satisfying to watch, both funny and educative. People just don't get what's truly beautiful I suppose.
I didn’t have a clue what was going on for most of it as I am currently in the 11th year of English secondary school (GCSE year) but you still managed to interest me with your enthusiasm and charisma. Basically what I’m saying is well fucking done and great videos🤙🏼
Second derivative I''(t) can be calculated through residues at (-i, +i) in Complex plane. Thus, I(t) = C1 + C2*t +pi/4 exp(-2t). If we estimate (sin(tx)/x)^2
You are actually giving my existence meaning with your videos. Thank you so much. I have been rewatching movies and animes, done maths exercises, and gamed a lot, but it really didn't help. Then I found your channel because of Tibees, and I must say, this definitely is the best cure for boredom.
Another clean way to do it is to note that the fourier transform of a triangle function from [-2,2] is sin^2(w)/w^2 and the fourier transform of e^(-|t|) is 1/(w^2+1), constants not withstanding. Having noted that, you can use Parseval's Identity to calculate the integral.
I don't know these are not taught at University/college ..this is so cool,Feynman technique,Laplace transform ,differential,integral all used up in 1 question
Absolutely amazing content, as always! :'D Your excitement when solving these problems really does shine through! A question though: Is it always true that the Dirichlet integral evaluates to pi/2, even with a factor t in the argument of the sine function? Anyways, a beautiful solution!
21:53 : "am i that bad at math?, why is there +pi/2 instead of -pi/2" After two more hours and a handmade demonstration: "Let's see what he obtains as a final result" 23:49: "oh... I see....." :)))) Amazing video bro! Keep up the good work! This really made my weekend!
I love Differential Equations and Laplace Transforms, they are so useful in Control Systems and Model Simulations. I was totally excited when you transformed the Integration into a initial value problem. Reminds me when I used integration to calculate an area, while my tutor was using an orthogonal approximation and was caught by surprise. The point is I was always afraid of math. Now, with content like yours, I turned my greatest fear into my greatest weapon. Thank you, senpai!
I realized a fact that if I replace t = 1 by t = -1, by observing the expression of I(t) I'm supposed to be getting the same result just because I(t) is an even function of t. But the expression of I(t) that we find out in the end is not an even function of t, but all those steps that we did for finding I(t) do not depend on the sign of t. And one thing that if we replace t by -t in the expression of I'(t) and I'''(t) (it doesn't affect on I''(t)), we will have a differential equation that doesn't have real eigenvalues.
Using Laplace transforms here was like cracking a nut with a sledgehammer. By inspection you can see the solutions for J(t) are either cosh(2t), sinh(2t) or exp(2t), exp(-2t) whichever you prefer. Then try a polynomial for the particular integral. Then balance constants. The LTs made me feel a bit sick
Great example of a solution by Feynman's technique! I got as excited as you in the video and started showing it to the first one I met! I showed it to my cat. He yawned and went back to sleep. 🙄
Great result but why did you use Laplace transform at 10:10 when you can just set y(t):=J(t)-pi/2 and obtain y''-4y=0 which solutions have the form y(t)=y(0) * cosh (2t) + y'(0)/2 * sinh (2t) ? (it's quicker and didin't need to know the theory of Laplace Transform ^^ ) (sorry for my bad english I'm a French student)
He did it like that because is his video and that's what he felt like doing. No matter how long or short it took or would of taken he felt like explaining it like that and that's what he did.
I mean he could have also used derivative operator and wrote (D^2 - 4)J=pi/2 (D-2)(D+2)J=pi/2 set y = (D+2)J (D-2)y=pi/2 and then solve it that way but as the other person said he likes doing the Laplace transform
A long and winding load to solve a linear differential equation with Laplace transform. Solve the characteristic equation of the LDE, and build the solution space of it. Then, some exponential functions will appear directly.
So perfectly explained. So easy to follow. So brilliant. So inspiring. Thank you Pi M and BPRP for this blessing. Possibly the Mozart of Math. Beautiful work boi.
Yes! I love your enthusiasm for maths :) That was a very wild ride (I haven’t learnt Laplace) but the ending was still very satisfying! You have a new subscriber~ ^_^
What a beast ! i didn't totally understand why you can use the partial derivative and the Laplacian transformation but to come up with that man .. chapeau like we say in my country !
I can like videos, but why can't I favourite it?? I think this is my favourite video I've ever seen. Showing it to literally everyone I know who vaguely is good at maths. NEVER STOP INTEGRATING!
I was so nervous when you made that sign error in the end, I was begging so hard that you notice it and I got so happy when you corrected it and got the correct answer 😂
I have another way of solving this differential equation. We have f"(x) = -2π + 4f(x) We add to both sides 2 f'(x) Then multiply be e^(2x) to bith sides and then we get. We actually do inverse of differentiation process (e^(2x) * f'(x) )' =( 2 *e^(2x)*f(x) - e^(2x) * π )' Therefore e^(2x)f'(x) - 2e^(2x)f(x) = -e(2x)*π + c1 We divide by e^(4x) both sides and we get ( f(x) ) ' ( ------ ) =( π/2 * e^(-2x) + ( e^(2x)) c1/4 * e^(-4x) + c2)' Hence f(x) = π/2 + c1/4 *e^(-2x) + c2*e^(2x)
Noticing that sin^2 (x) = 1/2*(1-cos (2x)) and that the integral is even you could also use the Cauchy theorem of complex analysis to obtain the same solution....but I naturally know that you had already thought about that! Anyway, congratulations for the brilliant solution!
Ugh I fucking love this guy. The exasperation when he looks into the camera knowing he has to do partial fractions. Such a labourious process that is so fucking easy to stuff up.
Lit af
blackpenredpen
Can u plz make a video of a different approach to this i integral😀i would like to see what you'll come up with❤
Could this integral be solved more simply by contour integration? I was thinking that you can break the (sinx)^2 into e^(2ix) + e^(-2ix) - 2 /(-4). Close in the upper-halfplane for the first piece, bottom half-plane for the second piece, and either direction for the third piece. Observe that the integral along the real number line from -infinty to 0 is the same as from 0 to +infinity (as the function is even)
Please make a video demonstrating contour integration, @blackpenredpen
Can you have the limits of integration be dependent on t as a consequence of using this technique?
Beyonce Queen you forgot the question mark ;)
Dank. It was so nerve racking watching you make those two sign mistakes. When you caught them it was like watching a bomb being defused.
Bruh I ain't even finished high school and I have no idea why this popped up on my recommendations, but I watched the whole thing
F
Same
He lost me at the Laplace stuff but he was too excited and I was too invested to stop watching
Me to
Same. I enjoyed it tho.
When you haven't learned Laplace transform yet but the symbols look cool
can confirm, laplace transforms are one of the coolest things you will learn
:)
Please discuss details clear picture of your work. Don't hurry
@@Jacob-ye7gu do u live in Pittsburgh?
Same pinch😇😇😇
who would win?
-3 green chalkboards full of maths
*-ONE MINUS SIGN BOI*
Xd
Well negative 3 chalkboards would be pretty useless for solving
@@yana_2_6_0 sounds like a challenge
@@milanstevic8424 i cant click read more
Using Laplace transforms for that is like bringing a shotgun to a game of paintball lol 😂
@@MAN_FROM_BEYOND I have always been told that, hehe
I agree. I solved it much more quickly using a standard method. By the way, the solution given on the vídeo is just one of the many solutions possible, wich appear naturally in the standard method.
Please make a video of the paintball solution 🙂
@@chrisjfox8715 it’s just a second order differential equation, quite easy to solve using standard method. Just look up on khan academy.
@@chrisjfox8715 I made one already, but in Portuguese.
I love how you can barely contain your enthusiasm.
This took me about 2 minutes. Complex analysis gives an extremely elegant solution.
Mark Saving Papa flammy likes to do it the cool way, without complex analysis. I bet its wayyyyyy easier with complex analysis (which I cant do yet so im forced to do it the cool way).
Do you have any recommendations for any online sources or books I could get to learn complex analysis? I havent been able to find a good one.
@@restitutororbis964 have you been able to find any? I'd be interested
@@categorille8330 Yes I got a book on complex analysis a month ago. Its called complex variables and applications by Ruel V. Churchill. I havent gotten that far into it but its pretty good for an introductory book in complex analysis from what I've seen so far.
Consult Papa Rudin.
Why use complex analysis if you can keep it real? See it as a training and the method he used was extremely cool and creative!
No one:
Absolutely no one:
Not even Ramanujan:
Papa Flammy: ok now we're gonna use the Laplace transform
"Not even Ramanujan" lmao
lol
“We’re going to use Leibniz rule”
*physics majors want to know your location*
gustavo espinoza i guess they call it feynman method 😬?
laventiny feynman TRICK
@@nombre3053 you meant METHOD
You used differential equations as an integration method... that's incredible. Laplace transforms at that! You beat Wolfram Alpha lol
Jordan Fischer Indeed. Laplace transforms are god like if you know when to use them and how to use them.
@alysdexia edgy
I honestly went super saiyan when the differential equation showed up. best integral ever
"The good thing is when doing partial fraction decomposition we just need to have the ability to read." - PapaFlammy
You are my God.
ok so I wanted to refresh myself on both Laplace transforms and Feynman's method and was about to look both up separately but I thought to myself I will watch one more meme integral video... yet here i find myself.
thank you
This is what I mean when I tell my friends that mathematics is beautiful. Results like these, those that just pop out of the blue and completely makes you unable to talk, those are the best moments you'll ever have. Beautifully done and indeed, a very beautiful integral which definitely beautified my Saturday evening =)
If those are the best moments you will ever have then you have one miserable ass life.
@@broadcast3ful
your comment got two likes over three years.
==> successful fail.
21:58 its amazing that you solved differential equation but missed the "minus" sign near the first term:
J(t) = -\frac \pi 2 e^{-2t} + \frac \pi 2
I'm putting off homework I need to do for homework I want to do. Your enthusiasm is what really makes this video great
What's also ridiculously awesome is that the graph of sin²(x)/(x²(x² + 1)) has a very close fit to the normal distribution curve or e^-(x^2)
the enthousiasm, the clickbait title, this is the best thing ever
If its real it isnt clickbait
There needs to be more people like you; salivating over Putnam problems alone is a sad life. I wish for a world where all RUclips thumbnails are as glorious as the one for this video
I absolutely love this video. This solution was absolutely ingenious , and the way everything simplified so nicely in the end weirdly pleases me. You actually did make my weekend, and as a result I am now a patreon. Keep up the incredible work!
I'm in highschool AP calc and this just shows how much more I can still learn
I can't understand why you're not a +1M subscribers channel, your content is so great, so satisfying to watch, both funny and educative. People just don't get what's truly beautiful I suppose.
You have the most infectious enthusiasm I've ever seen displayed. Subbed!
Loved your German accent. sounds like some Swiss.
I didn’t have a clue what was going on for most of it as I am currently in the 11th year of English secondary school (GCSE year) but you still managed to interest me with your enthusiasm and charisma. Basically what I’m saying is well fucking done and great videos🤙🏼
Second derivative I''(t) can be calculated through residues at (-i, +i) in Complex plane.
Thus, I(t) = C1 + C2*t +pi/4 exp(-2t).
If we estimate (sin(tx)/x)^2
I couldn't resist just solving I'' with complex analysis, so in the end it was faster. But it's nice to see those Laplace transforms once in a while.
Bro, you just got me so hyped to take differential equations. I love your enthusiasm!
You are actually giving my existence meaning with your videos. Thank you so much. I have been rewatching movies and animes, done maths exercises, and gamed a lot, but it really didn't help. Then I found your channel because of Tibees, and I must say, this definitely is the best cure for boredom.
Man, that thing inspired me, the way that the result just show up after that parametrization ❤️❤️❤️
Im glad you caught the minus sign error at the end. It was seriously bugging me.
Another clean way to do it is to note that the fourier transform of a triangle function from [-2,2] is sin^2(w)/w^2 and the fourier transform of e^(-|t|) is 1/(w^2+1), constants not withstanding. Having noted that, you can use Parseval's Identity to calculate the integral.
I don't know these are not taught at University/college ..this is so cool,Feynman technique,Laplace transform ,differential,integral all used up in 1 question
22:17 "But what's our initial condition?"
I just suddenly lost myself lol. We've come so far the shit looks nothing like the initial problem lmaoooo
I love how you love and enjoy Math, Papa.
One of the Greatest Video I recently watched, full of great things.
Thank you so much my *lovely Papa* ❤️
Awesome. Seriously, I just came home after giving a test. This legit made my day. This is one of the best question I saw in last few months.
Its 11 pm everyone is sleeping and the vid was on full volume..........
That intro....
Absolutely amazing content, as always! :'D Your excitement when solving these problems really does shine through! A question though: Is it always true that the Dirichlet integral evaluates to pi/2, even with a factor t in the argument of the sine function? Anyways, a beautiful solution!
If t is posivive, the Dirichlet integral is pi/2, if t is negative, it is -pi/2 :)
Andrei Popa Oh ok, thanks! :D
"We're using Laplace transforms!" (Throws chalk) hahahaha
Sin(y) = y^(0r+m*0*m)/g(a*y)
LanYarD N'(o)=u
21:53 : "am i that bad at math?, why is there +pi/2 instead of -pi/2"
After two more hours and a handmade demonstration: "Let's see what he obtains as a final result"
23:49: "oh... I see....."
:))))
Amazing video bro! Keep up the good work! This really made my weekend!
now i can die in peace
F
Lol
I love Differential Equations and Laplace Transforms, they are so useful in Control Systems and Model Simulations. I was totally excited when you transformed the Integration into a initial value problem. Reminds me when I used integration to calculate an area, while my tutor was using an orthogonal approximation and was caught by surprise. The point is I was always afraid of math. Now, with content like yours, I turned my greatest fear into my greatest weapon. Thank you, senpai!
I'm watching this now in a weekend nearly two years after the release and this still made my weekend better!
This method of using Laplace transform and Differentiating an integral got me so excited! When you threw your chalk, I threw my pen down.
I realized a fact that if I replace t = 1 by t = -1, by observing the expression of I(t) I'm supposed to be getting the same result just because I(t) is an even function of t. But the expression of I(t) that we find out in the end is not an even function of t, but all those steps that we did for finding I(t) do not depend on the sign of t. And one thing that if we replace t by -t in the expression of I'(t) and I'''(t) (it doesn't affect on I''(t)), we will have a differential equation that doesn't have real eigenvalues.
Using Laplace transforms here was like cracking a nut with a sledgehammer. By inspection you can see the solutions for J(t) are either cosh(2t), sinh(2t) or exp(2t), exp(-2t) whichever you prefer. Then try a polynomial for the particular integral. Then balance constants. The LTs made me feel a bit sick
I know papa probably won't see this but its wednessday and this still made my weekend. Overcomplicating can make shit easier
@@PapaFlammy69 lmao half my comment is rendered moot.
Nice work! This integral can also be calculated using complex analysis, with the residues theorem. I think it is much easier.
Great example of a solution by Feynman's technique!
I got as excited as you in the video and started showing it to the first one I met! I showed it to my cat. He yawned and went back to sleep. 🙄
Great result but why did you use Laplace transform at 10:10 when you can just set y(t):=J(t)-pi/2 and obtain y''-4y=0 which solutions have the form y(t)=y(0) * cosh (2t) + y'(0)/2 * sinh (2t) ? (it's quicker and didin't need to know the theory of Laplace Transform ^^ ) (sorry for my bad english I'm a French student)
He did it like that because is his video and that's what he felt like doing. No matter how long or short it took or would of taken he felt like explaining it like that and that's what he did.
I mean he could have also used derivative operator and wrote (D^2 - 4)J=pi/2
(D-2)(D+2)J=pi/2
set y = (D+2)J
(D-2)y=pi/2
and then solve it that way but as the other person said he likes doing the Laplace transform
A long and winding load to solve a linear differential equation with Laplace transform.
Solve the characteristic equation of the LDE, and build the solution space of it.
Then, some exponential functions will appear directly.
me two years ago in highschool: wow, this was really hard and cool
me first year in college: eAsY
'To your newborn son'
lol definitely
Great video. Really fun example of how solving an integral can be made easier if you can set it up as a diff eq. I solved with annihilators!
I am a litterature student now.
So perfectly explained. So easy to follow. So brilliant. So inspiring. Thank you Pi M and BPRP for this blessing. Possibly the Mozart of Math. Beautiful work boi.
I wish I could understand this better, it looks heavenly
You made my day! Amazing integral and approach. Really awesome! 🔥💪
Yes! I love your enthusiasm for maths :) That was a very wild ride (I haven’t learnt Laplace) but the ending was still very satisfying! You have a new subscriber~ ^_^
When you are tired, but then you realise that Flammable uploaded a 26 minutes long solution. I felt like Thanos
"This put a smile on my face"
What a beast ! i didn't totally understand why you can use the partial derivative and the Laplacian transformation but to come up with that man .. chapeau like we say in my country !
That was an awesome mashup of beautiful pieces of mathematics
at that triple derivative: and this is to go even further beyond!!! AAAAHHHHHH!!!
I don't have words to say what i think about this video. Just amazing
We all know that the cat is the thing RUclips recommended to you and it's what you came for
I can like videos, but why can't I favourite it?? I think this is my favourite video I've ever seen. Showing it to literally everyone I know who vaguely is good at maths. NEVER STOP INTEGRATING!
Nice job. Even Wolfram Alpha takes a second before coughing up the answer on this one.
:)
This technique is so beautiful.
This is f***ing unbelievable. Sheer genius.
Complex analysis! Residues! Ha ha ha.
But you are on fire.That is crazy.
I was so nervous when you made that sign error in the end, I was begging so hard that you notice it and I got so happy when you corrected it and got the correct answer 😂
I’m watching this two years late but it’s the weekend. How did he know???
I have another way of solving this differential equation.
We have
f"(x) = -2π + 4f(x)
We add to both sides 2 f'(x)
Then multiply be e^(2x) to bith sides and then we get.
We actually do inverse of differentiation process
(e^(2x) * f'(x) )' =( 2 *e^(2x)*f(x) - e^(2x) * π )'
Therefore
e^(2x)f'(x) - 2e^(2x)f(x) = -e(2x)*π + c1
We divide by e^(4x) both sides and we get
( f(x) ) '
( ------ ) =( π/2 * e^(-2x) +
( e^(2x)) c1/4 * e^(-4x) + c2)'
Hence
f(x) = π/2 + c1/4 *e^(-2x) + c2*e^(2x)
Any time a math video starts with “we are going to fuck shit up”, you know it’s gonna be a good time
This made my weekend better!
:)
mit deinem Enthusiasmus hast du dir mein Abonnement verdient^^hab jetzt richtig Bock mich nächstes semester in Integrationstheorie reinzusetzen, merci
:)
You certainly gave some light to my weekend
Way out there on the next planet. So many steps. But definitely flammable.
You did bring more light to my hot brazilian weekend. Congrats!
I tried the integral using complex analysis and the residue theorem which works well here as well
I don't know why I just found your channel, this shit is fuckin epic keep it up
Monstrous. Absolutely wild. Thank you father.
what a thing to watch on the weekend
I love your nerdy energy! Just freaking coollll!!!
=)
Awesome as always!
Minute 21:50 should be -pi/2 but you wrote +pi/2
Looking forward to the next video!
Differential Equation,Laplace Transform,Leibniz Rule all in one video!!! Awesome and Amazing!!
Wow..I just came back from Germany and this is the first thing I see on RUclips.. lit
You know you are excited when you throw your chalk.
I LOVE THIS OMG MY WEEKEND REALLY WAS MADE
Noticing that sin^2 (x) = 1/2*(1-cos (2x)) and that the integral is even you could also use the Cauchy theorem of complex analysis to obtain the same solution....but I naturally know that you had already thought about that! Anyway, congratulations for the brilliant solution!
Ugh I fucking love this guy. The exasperation when he looks into the camera knowing he has to do partial fractions. Such a labourious process that is so fucking easy to stuff up.
yeah, so relatable... but you gotta do what you gotta do :D
I discovered this channel today, this is **PURE GOLD**.
Definitely one of my favorite integrals of all time! Added to my cool math playlist
Thank you for the video! All of you friends are super awesome!
Never have I ever seen a Guy this excited over an integral
You save me from depression, thank you very much. It's insane
How did you think about differentiating that integral multiple times?! Blew my mind, and amazing video as always. I love your channel :-D
I wish my integral Calc prof had a German accent like yours. Would definately make lectures more enjoyable.