The Dirichlet Integral is destroyed by Feynman's Trick

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  • เผยแพร่เมื่อ 24 ธ.ค. 2024

ความคิดเห็น • 229

  • @LaughingManRa
    @LaughingManRa ปีที่แล้ว +648

    Feynman DESTROYS Dirichlet Integral with FACTS and LOGIC

    • @cristianv2850
      @cristianv2850 ปีที่แล้ว +1

      😢

    • @Juan_Carl0s
      @Juan_Carl0s ปีที่แล้ว +20

      *Leibniz

    • @cyrillechevallier7835
      @cyrillechevallier7835 ปีที่แล้ว +19

      Yeah but it relies on the fact that deriving an integral (with respect to a parameter) is the same as integrating the partial derivative :) which is a god level tool (and it’s hard to prove)

    • @orenawaerenyeager
      @orenawaerenyeager ปีที่แล้ว

      @@Juan_Carl0s plug the upper limit multiplied by differential of upper limit minus plug the lower limit multiplied by differential if lower limit

    • @kylethompson1379
      @kylethompson1379 ปีที่แล้ว +6

      @@Juan_Carl0s Thank you. This "Feynman technique" thing is obnoxious. Feynman was great enough on his own merits, he doesn't need any help.

  • @kdmdlo
    @kdmdlo ปีที่แล้ว +93

    Trefor: this trick requires that the order of differentiation and integration can be interchanged because your F'(s) is actually d/ds [ int_0^infty e^(-sx) sin(x)/x dx ] . You nonchalantly swapped the order of these two operations. This is thoroughly valid provided the integrand satisfies certain integrability conditions etc. Perhaps it would be worth noting this (and going over these conditions), so if people are looking to use Feynman's trick, they will be on the look out for potential tripping points.

  • @Sugarman96
    @Sugarman96 ปีที่แล้ว +100

    Since you briefly mentioned the Laplace transform, I feel like it'd be a waste not to mention the super important Fourier transform in this context, because the Fourier transform lets you solve the Dirichlet integral almost immediately. It turns out, the Fourier transform of a window function of from -1 to 1 is sin(w)/w, so using the inverse fourier transform, you get the value of the Dirichlet integral.

    • @donaldmcronald2331
      @donaldmcronald2331 ปีที่แล้ว +5

      I learnt both the fourier transform and reverse transform and immediately knew sine(x)/x. I wasn't aware of the link between them lol.
      I'd add that if you view sine(x)/x as a spectrum, its energy across actual time is indeed related to the integral of sine(x)/x from 0 to infinity.

    • @MrPoornakumar
      @MrPoornakumar ปีที่แล้ว +4

      @@donaldmcronald2331
      Your second sentence restates "Parseval" equation. It physically means, no matter in what domain (frequency or time) you integrate, the energy is same.

  • @NumbToons
    @NumbToons ปีที่แล้ว +6

    I just (today) learnt this integral in Fourier Transform, and here you come up with a video to make it permanent my memory.

  • @moritzberner8402
    @moritzberner8402 ปีที่แล้ว +5

    I directly tried to solve the integral of cos(x)/x with the same method and found out that this one diverges. Great video!

    • @smiley_1000
      @smiley_1000 ปีที่แล้ว +9

      The reason is the behavior as x tends to 0, where the function behaves approximately like 1/x. Perhaps you should consider the integral from 1 to infinity in order to get a more interesting result.

  • @Djenzh
    @Djenzh ปีที่แล้ว +95

    Beautiful! Funny that the Laplace transform shows up. I only knew how to do this integral by changing sin(x)/x into sin(xt)/x and then taking the Laplace transform of the entire thing, but your solution seems much easier :)

    • @DrTrefor
      @DrTrefor  ปีที่แล้ว +25

      I suppose this method is mostly equivalent. I'd actually suggest your is more true to the spirit of Feynman's trick and a bit more generalizable, but perhaps the way I showed slightly more efficient for 1st year calc students

    • @leif1075
      @leif1075 ปีที่แล้ว +2

      @@DrTrefor But why and how did Feynman or whoever else come up with this? I could never admit I couldn't or Im not a math whiz who would come up with this or something similar--why not just have e^x or e^sx where s is just a constant and make it a function of x still--was this tried--Thanks for sharing and hope to hear from you.

    • @zigajeglic3645
      @zigajeglic3645 6 วันที่ผ่านมา

      @@leif1075your integral still needs to converge so plugging in e^x wont do you any good

  • @cablethelarryguy
    @cablethelarryguy 4 หลายเดือนก่อน

    You have no idea how many videos I've watched about Feynmann's technique. I finally understand it. Thank you so much!

  • @emanuellandeholm5657
    @emanuellandeholm5657 ปีที่แล้ว +13

    I always love to watch different people's take on the Dirichlet integral. It's second only to the Gaussian integral for me. :)
    The interesting thing about the Dirichlet integral is that it's not Lebesgue-integrable. Put some of that stuff in your pipe and smoke it!
    Dr Bazett's take is basically a Laplace transform, and I think it's cute!

  • @giovanni1946
    @giovanni1946 ปีที่แล้ว +9

    You can't plug in s = 0 as the Feynman trick can only be applied with s > 0, due to the absolute convergence requirement, though the limit as s goes to 0 indeed equals pi/2

    • @miloweising9781
      @miloweising9781 ปีที่แล้ว

      Yeah you need to be careful when limiting to zero here. Probably there’s a nice dominated convergence argument to say that F is continuous at 0 from the right.

    • @xaxuser5033
      @xaxuser5033 ปีที่แล้ว

      @@miloweising9781 it is actually continus at x=0 and here is a complete proof of this fact: ( let f(s,x) be the function inside the integral )
      -for all x>0 the function s --> f(s,x) is continus.
      -for all s in R : x-->f(s,x) is continus.
      - there exist a continus integrable and positive function g : R+--> R+ such that for all x>0 and for all s in R we have :
      |f(s,x)|

    • @giovanni1946
      @giovanni1946 ปีที่แล้ว

      @@xaxuser5033 That's not correct, as e^(-sx) -> 1 when s -> 0, which is not integrable on R+

    • @xaxuser5033
      @xaxuser5033 ปีที่แล้ว

      @@giovanni1946 i didn't understand what u want to say , where did i write e^(-sx) ?

    • @giovanni1946
      @giovanni1946 ปีที่แล้ว

      @@xaxuser5033 This theorem cannot be applied here, you can't choose g(x) to be e^(-x) as e^(-sx) gets bigger when s -> 0

  • @flamitique7819
    @flamitique7819 ปีที่แล้ว +7

    Great video ! I just wanted to point out that even though the result is correct, the reasoning here wasn't completely true, or was at least incomplete : by this reasoning, which uses the dominated convergence theorem and it's equivalents to derivate under the integral, you can't directly prove this formula for all s greater or equal to zero, but the formula is only true for s stricly bigger than zero, meaning you can't directly plug in 0 at the end (this is because you can only dominate the first function you want to derive under the integral for all s>0). But the formula still holds for all s>0. So the correct way to prove it is to show that the limit as s goes to zero of F(s) is indeed the integral of sin(x)/x from 0 to infinity, and you can use the fact that the right side of the equation is continuous at zero to give the final result. However, showing that you can interchange the limit and the integral as s goes to zero is not that trivial, since you can't dominate the function properly. To do that, you first need to integrate by parts and only after that you can dominate the function properly and do an interchange of limits and integral that is valid, and get the final result.

    • @aua6330
      @aua6330 ปีที่แล้ว

      For the continuity at 0, you can also show that if you take the sequence of functions F_n(s) = integral from 0 to n of (...), the sequence converges uniformly on [0, infty(, and each F_n is continuous.

  • @ரக்ஷித்2007
    @ரக்ஷித்2007 3 หลายเดือนก่อน +1

    The beauty of Laplace transforms! Amazing video❤

  • @scollyer.tuition
    @scollyer.tuition ปีที่แล้ว +5

    The terminology related to this method is itself somewhat interesting - when I first came across it (about 40 years ago), I don't think it was given any specific name (except differentiating under the integral sign), then the name "Leibniz rule" seemed to become more popular, and in the last 5 years or so, the "Feynman method" began to reign supreme - a tribute to continuing popularity of Mr Feynman, I guess.
    And it's worth pointing out that there are conditions required for the method to work: continuity of f(x,s) in both x and s and (partial) df/ds over the region of integration, IIRC. (corrections gratefully accepted if I misremember).
    Also, there's a generalisation of the method that takes account of variable limits depending on s.

    • @frenchimp
      @frenchimp ปีที่แล้ว +1

      I guess Feynman's contribution to the Leibniz method was "to hell with boring mathematical justifications".

    • @radekvecerka1115
      @radekvecerka1115 ปีที่แล้ว +2

      there are 4 conditions
      1)continuity (or more generaly the function has to messurable)
      2)continuos partial derivatives
      3)very important is you have to find a mayorant to derivatice of the intagrated function with respect to the parameter, which must have finite value when integrated
      4)you have to find a least 1 parameter for which you can calculate the integral
      Overall the most important part about this ''trick'' is figuring out whether you can even use it!

  • @purplenanite
    @purplenanite ปีที่แล้ว +7

    That's a neat trick!
    i came across this integral yesterday, interestingly enough - although now I know how to do it faster!

    • @DrTrefor
      @DrTrefor  ปีที่แล้ว +4

      Isn't it cool!?

    • @leif1075
      @leif1075 ปีที่แล้ว

      @@DrTrefor I don't see why you take the limit at 5:52 instead of say plugging in the value of zero for s to get C? Isn't that more logical and intuitive?

    • @leif1075
      @leif1075 ปีที่แล้ว

      @@DrTrefor and I don't see why anyone would take the limit as S goes to infinity--isn't there some other way to get C--if you set s to zero you get F(0)=C

    • @MichaelRothwell1
      @MichaelRothwell1 9 วันที่ผ่านมา

      ​@@leif1075Unfortunately that doesn't work here as it's the value at s=0 we're trying to find.

  • @ரக்ஷித்2007
    @ரக்ஷித்2007 3 หลายเดือนก่อน +1

    3:10 This can also be solved using complex numbers by rewriting this as Im(e^(-sx-ix)).

  • @johnchessant3012
    @johnchessant3012 ปีที่แล้ว +19

    Fun fact: the integral of sin(x)^2/x^2 from 0 to infinity is also pi/2

    • @Ninja20704
      @Ninja20704 ปีที่แล้ว +9

      But go to the third power and then it breaks down. That integral will be 3*pi/8

    • @NumbToons
      @NumbToons ปีที่แล้ว +3

      wow

    • @OmegaQuark
      @OmegaQuark ปีที่แล้ว

      ​@@Ninja20704 Is there a generalized closed formula for the n-th power of sin(x)/x? Like with the Dirichlet series, or the Zeta function for the even positive integers

    • @violintegral
      @violintegral ปีที่แล้ว

      @@OmegaQuarkyes, but it's a bit complicated

    • @Ninja20704
      @Ninja20704 ปีที่แล้ว

      @@OmegaQuark for the zeta function at even positive integers, its been proven that zeta(2n) will always be some rational multiple of pi^2n. Figuring out that rational multiple is pretty complicated but doable.

  • @laydenhalcomb4559
    @laydenhalcomb4559 4 หลายเดือนก่อน +1

    I always love your videos Dr. B. Thank you for sharing. I’m now going to utilize the Feynman trick in my calc 3 class

  • @adw1z
    @adw1z 6 หลายเดือนก่อน

    Another way:
    I = Im[int(0,inf) e^(iz)/z dz]
    J = int(0,inf) e^(iz)/z dz
    Draw a semicircular contour of radius R in the top right quadrant of the complex plane, which goes around the simple pole at z = 0.
    The bottom contour -> J as r->inf.
    By the indentation lemma, the contribution around z = 0 pole is -i*pi/2 as epsilon goes to 0, where epsilon is the distance away from the pole of the contour going around 0.
    The contour from (infinity)i to 0 gives 0 contribution, as the integrand tends to 0.
    The circular contour joining R and iR tends to 0 as R tends to infinity by Jordan’s Lemma.
    The full contour contribution is 0 since it encloses no singularities. Thus, we have:
    J - i*pi/2 = 0
    ==> J = i*pi/2
    ==> I = int(0,infinity) sin(x)/x dx = pi/2
    N.B sin(x)/x has a removable singularity at x = 0, and hence the integral converges. The same cannot be said for cos(x)/x; that integral diverges.

  • @samueldeandrade8535
    @samueldeandrade8535 ปีที่แล้ว +3

    We live at a weird time. When doctors make videos with clickbaity titles. *Roll eyes*.

  • @ketankyadar5228
    @ketankyadar5228 ปีที่แล้ว +3

    I am really happy to see Differentiation Under Integral sign rule here to calculate integration of Sinx/x.
    Actually today in the class i taught this rule and after that i saw your video.
    I amazed that how you start with combining exponential term in the integral, In real life there are so many situations where you can use this cause there exist always parameter with your function.

  • @peterpan1886
    @peterpan1886 ปีที่แล้ว

    You can also view the integral as the imaginary part of the integral of e^(izx)/x, evaluated at z=1. Now integrating under the integral sign yields I'(z)=[e^(izx)/z]_0^\infty = -1/z for complex z with positive imaginary part. Hence I(z)=c-log(z). Remember that we are only interested in the imaginary part of I(z), therefore we only need the imaginary part of c. Now let the imaginary part of z approach infinity, while the real part remains constant. Then the imaginary part of log(z) will approach pi/2, while the integral of e^(izx)/x approaches zero. Hence Im(c)=pi/2 and the limit of Im(c-log(z)) as z approaches 1 is pi/2.

  • @ShanBojack
    @ShanBojack ปีที่แล้ว +4

    You remind me of my tuition teacher who is also a big mathemagician and you both are my ideals 🙌

    • @DrTrefor
      @DrTrefor  ปีที่แล้ว +3

      So kind!

  • @andrewharrison8436
    @andrewharrison8436 ปีที่แล้ว +1

    Some bits of maths are like p v np, really hard to first find the method (np), but manageable to verify that the solution works (p). Actually it isn't "some bits", it is a lot of the bits and it is cumulative - Newton's "If I have seen further than other men it is by standing on the shoulders of giants".

  • @David-dvr
    @David-dvr ปีที่แล้ว +5

    Although this trick is named after Feynman, I believe he found it in an advanced calculus book his high school physics teacher gave him. It was apparently developed, at least partially, by Leibniz:
    “I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. [It] showed how to differentiate parameters under the integral sign - it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. [If] guys at MIT or Princeton had trouble doing a certain integral, [then] I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me.” (Surely you’re Joking, Mr. Feynman!)
    www.cantorsparadise.com/richard-feynmans-integral-trick-e7afae85e25c

    • @DrTrefor
      @DrTrefor  ปีที่แล้ว +2

      Cool!

    • @andrewharrison8436
      @andrewharrison8436 ปีที่แล้ว

      Must reread my Feynman books - nice shout out by Feynman to his teacher.

  • @moeberry8226
    @moeberry8226 ปีที่แล้ว +3

    It’s more common to think that when applying Feynmans trick that you would put the parameter s inside of the sin(x) function aka sin(sx). The only reason that you use this decreasing exponential function is that so it converges on the interval 0 to infinity because the original integral converges as well. If you use the first method then you will obtain cos(sx) after differentiation with respect to s. Which does not converge. You should explain it this way to the students on TH-cam.

    • @Ayush-yj5qv
      @Ayush-yj5qv 10 หลายเดือนก่อน

      true when i did this method i end up getting sin of infinite which i dont even know

    • @M.Z.M.N.
      @M.Z.M.N. หลายเดือนก่อน

      Thanks mr,obvious

    • @moeberry8226
      @moeberry8226 หลายเดือนก่อน

      @ that’s not obvious one bit.

  • @quantizedMind
    @quantizedMind 2 หลายเดือนก่อน

    7:14 you can only use s>0 because then in 3:18 left part doesn't disappear.

  • @roderictaylor
    @roderictaylor 6 วันที่ผ่านมา

    Chapter 17 of Vladimir A. Zorich’s Mathematical Analysis II goes into detail justifying both differentiating under the integral sign for s>0 and that F(s) is continuous at s=0.

  • @cameronspalding9792
    @cameronspalding9792 ปีที่แล้ว +1

    @ 3:31 Personally I would have used the formula for sine in terms of complex exponentials

  • @FayçalDaoudi-n1z
    @FayçalDaoudi-n1z ปีที่แล้ว +2

    Is there a video for when it is allowed to change the order of an integral sign followed by a summation notation? Thank you.

    • @DrTrefor
      @DrTrefor  ปีที่แล้ว +1

      I really should make this video

  • @General12th
    @General12th ปีที่แล้ว +3

    Hi Dr. Bazett!
    Feynman's technique is one of my favorite in all of integral math.

    • @frenchimp
      @frenchimp ปีที่แล้ว +2

      So long as you are aware it owes nothing to Feynman.

    • @General12th
      @General12th ปีที่แล้ว

      @@frenchimp No. I'm not aware of that. Why don't you explain it to me?

    • @kylethompson1379
      @kylethompson1379 ปีที่แล้ว +1

      @@General12th Feynman is on video on youtube saying how he read it in a book, and anyway it's all just based on work done by Leibniz 100s of years ago. Feynman did re-popularise it though.

  • @ulyssesfewl1059
    @ulyssesfewl1059 ปีที่แล้ว

    Did I miss something? At 1:29 if you set s = 0, you do not get back what you started with, since the entire exponent is zero if s = 0, and e^0 = 1.

  • @namelastname2244
    @namelastname2244 ปีที่แล้ว

    3:22 If s is 0 and not a positive, then why do you consider it a negative exponential?

  • @utuberaj60
    @utuberaj60 ปีที่แล้ว +1

    Hi Dr Brazet. Wonderful video.
    But, I have a question here. Why is this method known as Feynman's technique? In fact the idea of Differentiating Under the Integral sign (DIUS) was due to Leibnitz, and also the idea of introducing a parameter 's' in the form e^-sx is the Laplace transform or can even be thought of as the Gamma function?
    Why do you then still call this the Feynman trick? Just wanted to know.

    • @itellyouforfree7238
      @itellyouforfree7238 ปีที่แล้ว

      Because at least in the physical community in the USA it has been popularized by Feynman

    • @anuragkr3026
      @anuragkr3026 11 หลายเดือนก่อน

      Sometimes good thing are better not to be shared.

  • @brianneill4376
    @brianneill4376 ปีที่แล้ว

    Pi devided by 2 amoubts to 1/3 of a Cubical measure.
    All measures are 3D so must be no more or less than Cubical or "Powered to 3", before and or after that are simply sizes that are fractions, Positive or negative to the control body size.

  • @jinoobaek
    @jinoobaek 5 หลายเดือนก่อน

    Thanks!

    • @DrTrefor
      @DrTrefor  5 หลายเดือนก่อน

      Thank you so much!!

  • @vpambs1pt
    @vpambs1pt ปีที่แล้ว +1

    fuun, so the are under sin(x)/x over R is pi.
    This function is just, the typical sinusoidal sin(x), but as the x increases, it is divided by the factor x, so each period 2pi, it's just the sin wave getting "linearly" smaller.
    On the other hand, this scalling over each x, makes it such that the area over the whole domain is pi, which is the area of a circle of r=1.
    Which clearly has some meaning, the sin is constantly alternating its sign, and x is either x>0 or x

  • @jeremytimothy3646
    @jeremytimothy3646 ปีที่แล้ว +2

    This is just spiced up Laplace transforms 😂. Nice video though always happy to learn abit more math.

    • @anuragkr3026
      @anuragkr3026 11 หลายเดือนก่อน

      I dont anymore people to know about it ,
      Sometimes good things are not good to be shared ❤

  • @maroc4747
    @maroc4747 ปีที่แล้ว +2

    Great explanation. My follow up question would be, why does this work and when should one use this trick? I only knew the Double integral solution for this Problem.

    • @DrTrefor
      @DrTrefor  ปีที่แล้ว +1

      With a lot of these integration tricks, ultimately it works when it works. With some practice you can gain some intuition for when you can parameterize and integral and do this differentiate with respect to the parameter trick, but there isn't some general rule for when it always works.

  • @xl000
    @xl000 ปีที่แล้ว +1

    it must be weird to look and point at a green screen when saying "Like this one" at 0:04

  • @fordtimelord8673
    @fordtimelord8673 ปีที่แล้ว +2

    Contour integration works great for this problem too. Using a semicircle contour and taking the imaginary part of the integral of e^iz/z is maybe even an easier method.

    • @revanth36
      @revanth36 ปีที่แล้ว +1

      Interestingly yes😀

  • @attiladalmadi182
    @attiladalmadi182 4 หลายเดือนก่อน

    Dirichlet Integral, good explanation

  • @madhavpr
    @madhavpr 8 หลายเดือนก่อน

    Pretty cool trick. I'm okay with all the steps except perhaps the interchange of limit and the integral as s->infinity. Is there some theorem in real analysis that justifies this?

  • @hvok99
    @hvok99 ปีที่แล้ว +1

    Wow that was so satisfying.

  • @fatemekashkouie3662
    @fatemekashkouie3662 ปีที่แล้ว +2

    Hello Dr.Trefor Bazzet,
    I wanted to take a moment to thank you for all the beautiful content you're creating. They're awesome.
    By the way, I'm supposed to do one of my class projects using maple. But I haven't gotten used to it. Do you suggest any particular tutorial teaching how to use maple?

    • @DrTrefor
      @DrTrefor  ปีที่แล้ว +3

      Thank you! Sorry I don’t have any particularly great resources at my fingertips, mostly because it can do so much it really depends what you need to use it for!

    • @fatemekashkouie3662
      @fatemekashkouie3662 ปีที่แล้ว

      @@DrTrefor I am supposed to solve differential equations for a dynamical systems course.
      Anyways, I think if I google every step, it will be somehow manageable.

  • @journeymantraveller3338
    @journeymantraveller3338 ปีที่แล้ว +3

    Isn't Feynman's trick just using Liebniz's Rule on the Laplace transform? Also, the DI method works well for these IBPs .

  • @Grassmpl
    @Grassmpl ปีที่แล้ว +1

    Doesn't feymann trick needs some sort of uniform convergence?

  • @iamreallybadatphysics
    @iamreallybadatphysics ปีที่แล้ว +1

    Thank you so much Dr. Trefor

    • @DrTrefor
      @DrTrefor  ปีที่แล้ว +1

      You're most welcome!

  • @ProfeJulianMacias
    @ProfeJulianMacias ปีที่แล้ว +1

    Excellent Math problem

  • @airsquid8532
    @airsquid8532 ปีที่แล้ว +3

    Always a good day when Dr trefor bazette shows up in my TH-cam recommended

  • @sambhavgupta4653
    @sambhavgupta4653 ปีที่แล้ว +2

    Can't we use exponential definition of sine, (exp(ix)-exp(-ix))/2i to solve it?

    • @DrTrefor
      @DrTrefor  ปีที่แล้ว +3

      This can replace part of the computation (the double integration by parts) but you still need to use some trick at the beginning before you can use this.

    • @sambhavgupta4653
      @sambhavgupta4653 ปีที่แล้ว

      Thanks! I'll try.

    • @mokouf3
      @mokouf3 ปีที่แล้ว

      And you use the same Laplace Transform / Feynman's Technique at start right?
      If yes, I recommend you use these rules instead:
      sin(x) = Im(e^xi)
      k * Im(z) = Im(kz)
      ∫ Im(z) dx = Im(∫ z dx)

    • @adamc973
      @adamc973 ปีที่แล้ว

      Easier to note that this is the imaginary part of e^(iz)/z over the interval [-infty,infty] in the complex plane, and work over a semi-circle contour in the upper half plane.

  • @maxp3141
    @maxp3141 ปีที่แล้ว +2

    Nice, I think I would had inserted sin(x) = (e^ix - e^-ix) instead of integrating by parts, but I’m not sure if that makes things easier or not..

    • @active285
      @active285 ปีที่แล้ว

      Yes similarly! Using the Feynman trick here reflects the relation to complex analysis: Choosing the function f(z) = exp(iz)/z and integrate over the (positive) indented semicircle and some appropriate contour avoiding 0.

  • @soufianehrilla5533
    @soufianehrilla5533 ปีที่แล้ว +1

    C'est juste très élégant

  • @dfcastro
    @dfcastro ปีที่แล้ว

    What if instead of during the integration by parts you apply again the Feymann technique and than (probably) will get a differential equation and solve it?
    Would it work?

  • @realdbcooper3423
    @realdbcooper3423 4 หลายเดือนก่อน

    Would that mean integrating six/x without limits give us pi? Im very interested in using feynmans trick cause I currently in the progress of learning Fourier analysis.

  • @MushookieMan
    @MushookieMan ปีที่แล้ว +1

    I thought the Laplace transform required a complex 's'. Does that affect this method?

    • @DrTrefor
      @DrTrefor  ปีที่แล้ว

      It can be complex, but we ultimately are only using it to get this differential equation which we only evaluate at infinity and zero so it doesn’t matter if other values could be complex

    • @mokouf3
      @mokouf3 ปีที่แล้ว

      Remember real number set is a subset of complex number set.
      If an equation is valid in the whole complex plane, the same equation is valid in the whole real number line, all we need is to limit the imaginary part to 0.

  • @duckyoutube6318
    @duckyoutube6318 ปีที่แล้ว +1

    What doesnt Feynman destroy? Guy was a beast.

  • @LeviDanielBarnes
    @LeviDanielBarnes 6 หลายเดือนก่อน

    But (3:25) if s=0, the exponential doesn't kill the cos term. This is not strictly valid for the value of s we care about.

  • @RobinTester
    @RobinTester ปีที่แล้ว

    Complex analysis ON TOP

  • @saidfalah4180
    @saidfalah4180 11 หลายเดือนก่อน

    Can we use (sin x/x) / pi/2 in probability ?

  • @xiangbocai-ns4sg
    @xiangbocai-ns4sg ปีที่แล้ว +1

    What a fantastic method

    • @DrTrefor
      @DrTrefor  ปีที่แล้ว +1

      Glad you think so!

  • @Sheelam123
    @Sheelam123 11 หลายเดือนก่อน

    Can we apply cauchy integral test here??

  • @speedbird7587
    @speedbird7587 ปีที่แล้ว +1

    Hello Professor,
    Thanks for all your brilliant videos,
    It was a really nice and technical( a bit similar to laplace technique)
    although I couldn't find a proof or nonexample for this technique from the internet,
    I am very curious to know that could we use this technique by any function other than exponential terms, or is it because of the uniform continuity of the laplace transform that we can use this trick?
    since the problem can also be thought as a differential equation/ a dynamical system biforcating on the parameter s .
    I really would like to know more about it.
    Thank you.
    Br,

  • @padraiggluck2980
    @padraiggluck2980 11 หลายเดือนก่อน

    When I listen to a Feynman lecture I hear a smart version of Ed Norton.

  • @edoardoferretti5493
    @edoardoferretti5493 ปีที่แล้ว

    What is the limit of F(s) as s goes to - infinity? Doesn't the relation we found fail since the integral is not bounded, while arctan is? How do I define the domain of validity of the identity?

    • @reeeeeplease1178
      @reeeeeplease1178 ปีที่แล้ว

      It may be that the step at 1:47 only works for positive s. What we are doing is interchanging 2 limits (pulling the derivative inside the integral), which we can't *freely* do

  • @urnoob5528
    @urnoob5528 ปีที่แล้ว +2

    damn this is basically a laplace transform trick :O
    wat a coincidence that im learning laplace transform right now

    • @DrTrefor
      @DrTrefor  ปีที่แล้ว +1

      Laplace transform is SO useful!

  • @thinkacademy8377
    @thinkacademy8377 ปีที่แล้ว +1

    I don't know why this is called Feynman's trick? differentiation under the integral sign is known before Feynman. It's just the application of Leibniz theorem for integrals dependent on a parameter!

    • @DrTrefor
      @DrTrefor  ปีที่แล้ว

      Also, the types of integrals Feynman was considering were nothing like these, but nevertheless the attribution seems to stick

    • @frenchimp
      @frenchimp ปีที่แล้ว +1

      @@DrTrefor So why perpetuate the ridiculous notion that this technique is due to Feynman in these videos?

  • @md.musaal-kazim1774
    @md.musaal-kazim1774 10 หลายเดือนก่อน

    we can use laplace transformation?

  • @roderictaylor
    @roderictaylor 23 วันที่ผ่านมา

    To justify the answer, we need to show that F(s) is continuous at s=0. Our integrand is not dominated in absolute value by an integrable function for s near 0, so this cannot be justified using the usual argument. I was able to do it, but it took me quite a while. Even so, I'm still not intuitively sure why it works.

  • @techsolabacademy9980
    @techsolabacademy9980 ปีที่แล้ว +2

    Dr. You are amazing❤

    • @DrTrefor
      @DrTrefor  ปีที่แล้ว +1

      Thank you!

  • @anietiethompson5375
    @anietiethompson5375 9 หลายเดือนก่อน

    Please which app are you using to plot those functions?

  • @Ron_DeForest
    @Ron_DeForest ปีที่แล้ว

    I’m sure this is beneath your notice but I have to ask and hope for the best. I need to know how to find a point in 3 space that’s equidistant from 3 other points. I’ve been looking online and for the life of my I can’t find how to do it. It’s been a very long time since I took any math classes.

  • @EVERYTHING_FACTORY
    @EVERYTHING_FACTORY ปีที่แล้ว

    Please make a mathematics books recommendation video

  • @robertcrompton2733
    @robertcrompton2733 ปีที่แล้ว +1

    Wow! Loved it!

  • @bobdavisbeta
    @bobdavisbeta ปีที่แล้ว

    All the physicists think that this method is Feynman trick but it existed way way before. It is a little abusive to atribute the credit to Feynman when this is actually extremely classical.

    • @Amsterdampardoc1
      @Amsterdampardoc1 ปีที่แล้ว

      Every physicist knows that the integration technique existed before Feynman, it’s just kind of that Feynman was famous for using it.

  • @chunghimchu3313
    @chunghimchu3313 7 หลายเดือนก่อน

    arctan (♾️) not only equal to π/2 , it can also be 5π/2, how do you prove that π/2 is an unique answer?

  • @amauta5
    @amauta5 ปีที่แล้ว

    Is this the only way to solve the original integral?

  • @pygmalionsrobot1896
    @pygmalionsrobot1896 ปีที่แล้ว

    Brilliant !! Thanks for making this video :D !!!

  • @Toxic__rl
    @Toxic__rl ปีที่แล้ว

    any smart way to solve int of x^100 sinx dx?

  • @Volk715
    @Volk715 ปีที่แล้ว

    Well, Feynman was the first "hacker" in history...I'm not surprised about his ability to tackle dirichelet integrals..

  • @hippospudweb
    @hippospudweb ปีที่แล้ว +1

    Where can I get that t-shirt?

    • @DrTrefor
      @DrTrefor  ปีที่แล้ว +1

      Merch link in description!

  • @marciliocarneiro
    @marciliocarneiro ปีที่แล้ว

    If we expand sin(x) in Taylor´s series and divide by x we obtain arctg(x) in just one step and the answer much more fast

  •  11 หลายเดือนก่อน

    Muy buen video Genio!

  • @mr_angry_kiddo2560
    @mr_angry_kiddo2560 ปีที่แล้ว +1

    Will you give a proof for Existence of local Maxima b/w two consecutive local minima😢

    • @MichaelRothwell1
      @MichaelRothwell1 9 วันที่ผ่านมา

      You need your function to be continuous, of course. Then the Weierstrass Extreme value theorem does it for you. See in Wikipedia.

  • @alphalunamare
    @alphalunamare ปีที่แล้ว

    Is 2:06 even legal? I suppose it must work in most common cases but can you prove it generally? I guess you can't and it is indeed strange. Anyway I really enjoyed that 🙂 It's a bit like with platonic solids, you differentiate its Volume formula and get its Surface Area formula ... and most people are convinced ...except that it only works for Platonic Solids not the general mish mash you'll meet in Real Physics.

  • @Helibenone
    @Helibenone ปีที่แล้ว

    I love your vids could you do some on statistics and probability

  • @kina4288
    @kina4288 3 หลายเดือนก่อน

    Look like the total area of this function equals to pi, in other words, the area of a unit circle. Contrast to the Gaussian bell shape curve where the square of the total area is that of a unit circle. Very interesting, no wonder people argue that god doesn’t play dice.

  • @kaiserali5928
    @kaiserali5928 ปีที่แล้ว

    Sir I have a question to you. I am an Engineering student from Bangladesh. Sir I derived an alternative procedure or technique to solve a math content which is being solved in complex ways nowadays. Now I want to publish it. Can I use LaTeX to write my paper? Again how and where I can publish it.
    If you give me some piece of advice then it will be very helpful for me.

    • @JuanRomero-re4qz
      @JuanRomero-re4qz ปีที่แล้ว

      Perdón!
      Busca una revista científica de tú localidad, y midiendo el terreno puedes pasar a una publicación en idioma ingles.

  • @domc3743
    @domc3743 ปีที่แล้ว +4

    brother didnt mention why we can switch order of differentiation and integration, Leibniz rule applies because F(s) bounded and continuous

    • @DrTrefor
      @DrTrefor  ปีที่แล้ว +3

      Oh thanks, I actually had it in my notes to point this out verbally but somehow forgot to say it out loud during recording time:)

    • @ShanBojack
      @ShanBojack ปีที่แล้ว +1

      ​@@DrTrefor can you elaborate it to me please sir

  • @otterlyso
    @otterlyso 4 หลายเดือนก่อน

    This is Leibniz's integral rule. More context here: en.wikipedia.org/wiki/Leibniz_integral_rule

  • @olegzubelewicz3604
    @olegzubelewicz3604 ปีที่แล้ว

    Mr. Feynman thought that math began with him. 🤣

  • @puh8825
    @puh8825 ปีที่แล้ว +1

    That's insane

  • @adityavsx
    @adityavsx 2 หลายเดือนก่อน

    whthe can’t we do sin(x)= sin(sx) instead?

  • @imad5206
    @imad5206 11 หลายเดือนก่อน

    good prof. thankks

  • @radmir_khusnutdinov
    @radmir_khusnutdinov ปีที่แล้ว

    I believe using the residue is the fastest way to calc that integral

  • @maths_505
    @maths_505 ปีที่แล้ว

    The Dirichlet integral is AWESOME!
    I made a video on solving it using 5 different ways including Feynman's trick:
    th-cam.com/video/MuP6NtGwLTk/w-d-xo.html

    • @daddy_myers
      @daddy_myers ปีที่แล้ว +1

      Damn. Imagine posting unsolicited ads for your channel.

    • @maths_505
      @maths_505 ปีที่แล้ว

      @@daddy_myers 😂😂😂
      Who cares
      I'm the one who did a marathon on it.
      You think I wouldn't be hyping it 😂😂😂

  • @davidwright5719
    @davidwright5719 ปีที่แล้ว

    Argh, don’t use double integration by parts here. Just use sin(x) = (e^{ix} - e^{-ix})/(2i) to get an integral involving only exponentials. This is almost always the best way to do almost any calculus problem involving trig functions. (Also non-calculus problems. Never worry about remembering a double-angle formula again.)

    • @piyusharora5327
      @piyusharora5327 ปีที่แล้ว

      Our professor taught like what you are saying when he was teaching the same thing in video in the name of Laplace transform.

  • @shivamkardam4608
    @shivamkardam4608 ปีที่แล้ว +1

    Amazing sir 😮😮😮😮😮

  • @baptistebermond2082
    @baptistebermond2082 ปีที่แล้ว

    well it is beautiful, but the real question is can we define the function as continuous and derivable everywhere which makes the trick less evident

  • @zegrirsaid2855
    @zegrirsaid2855 ปีที่แล้ว

    thank you

  • @akerosgaming7400
    @akerosgaming7400 ปีที่แล้ว

    I don't think the conditions needed to say that F is differentiable and take the partial differential inside the integral are there.

    • @roderictaylor
      @roderictaylor 6 วันที่ผ่านมา

      Chapter 17 of Vladimir A. Zorich’s Mathematical Analysis II goes into detail justifying both differentiating under the integral sign for s>0 and that F(s) is continuous at s=0.

  • @maksgo498
    @maksgo498 ปีที่แล้ว

    I would use taylor