when Feynman's integration trick doesn't work... 

Yep! Feynman credited Fredrick Wood's text book "Advanced Calculus" for where he learned this, and, sure enough, Woods shows the function must be continuous to switch differentiation and integration.

I love Feyman too! but please don't say this is his integration rule. He only popularized it among engineers. This rule was well known by mathematicians since Leibniz time.

In front of the integral it would be better to write d/dx rather than the partial derivative because we are indeed dealing with a standard derivative in x of a function of the variable x

Great video ! Altough I've seen several videos using Feyman's trick to solve the first integral, I've never seen one that was correct, because there is an really important step you forgot to mention to solve the first integral. With this reasoning, which uses the dominated convergence theorem and is equivalent 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.

Hi Dr. Penn! I've never seen the integral of sin(x)/x evaluated in quite that way, at least not with those specifics.

I always thought of Feynman's technique as introducing the extra variable so you have something to switch with Leibnitz's Rule, rather than any other application of that rule.

Could you make a video about the validity of changing either integration or differentiation with the summation sign!

At 6:22: Professor Pen: - Here we have our "DU" ear muffs. Me: - I understood that reference!

This method is NOT called "Feynman Integration" , IT'S CALLED *Leibniz Integral Rule* . Gottfried Leibniz DISCOVERED THE RULE, Feynman POPULARISED IT. THIS IS Leibniz's technique, NOT FEYNMAN'S. GIVE THE CREDIT TO THE RIGHT PERSON FOR GOODNESS SAKE 😡😡

3:26 Is it not a problem that in order to have exp(y(-x+i)) approached zero when y approaches inf ,the x value must be strictly positive (if x =0 , the value of the limit will be undefined circling around origin of the complex plane)? Because eventually we're interested in x =0.

its cute but most, even engineers will simply choose to use transform methods.

f is interesting. In polar coordinates (for r not 0), f is independent of θ. Indeed, f(r,θ) = cos^3(θ) sin(θ)

In complex analysis, functions need to be holomorphic in the defined region to use certain techniques. But as a result, they're well-behaved and infinitely differentiable.

I haven't learnt Complex Analysis, can anyone tell me about why taking the imaginary part to the front works? Just some intuition would be enough

I dont see why exp(iy - xy) -> 0 as y -> inf

You can that integral via Feynman, but you need to express your integrals in terms of the correct distributions

Feynman's mathematics was a bag of tricks to solve problems.

Feynman got the integration trick from the book "Advanced Calculus" by Woods.

I need some help to understand. At 1:00, he's adding an exponential in the integral. Is there some book or some video somewhere that explain this trick with more details and formalism?

Michael always chooses interesting content.

@3:22 why does y approaching infinity make the numerator 0?

Is there a specific restriction to be imposed on ( not only f) the partial derivative , both ( fx and fy)continuous functions?

At 9:25, importantly that is not a continuous function either just like the original f. Along the path x = y/2, it blows up to infinity and along x=y it equals 0. It shouldn't be surprising at all that a non-continuous function is also not differentiable and this is the concrete reason the trick does not work in this case.

This happens because the integral has to be linear in the derivative variable to Leibniz Regel to appy

3:25 as y approaches infinity e^ky will approach infinity not zero

Will this work if f(x,y) is defined = 1/2 at (0,0)?

Nice video :)

3:57 how is the numerator 0 when you have e^i*inf?

Does it always work when f is continuous

anyone know if there is a connection between the schwarz lemma and lie algebras?

UConn represent!

Very nice and well explained. Thank you!

Thank you, professor.

Statistics problem Mrs. Smith lives in a rural area. Each day she brings produce into town at 4 a.m. There's one traffic light in town with red or green every 30 seconds. She turns a corner that reveals the traffic light within 30 seconds of travel at the posted speed limit. She wishes to calculate a strategy that will maximize her average velocity going through the light over time. There's almost never any other traffic at that time of the morning, so any strategy can be used.

Awesome, again! What's the difference between the starting equation and expressing the fundamental theorem of analysis as a definite integral of Newton's heat equation? I suspect the implications could be relevant to the recent, more entropy oriented formulations of mechanics; often leaning on Noether current conservations.

Feynman stole the method from an unknown person called leibniz and just say he read it somewhere. Everything is leibniz here

Funny enough, so far no one noticed that Leibniz's Rule, as it is shown in the video, can be only applied when the domain of integration is compact. So we need some extension of this rule if we want to compute integrals on unlimited domains of integration. Probably, what is really involved is some sort of application of Lebesgue's theorem of the dominated convergence.