Differentiation Under Integral Sign (2015) [pdf]

https://web.williams.edu/Mathematics/lg5/Feynman.pdf

101 59 5 years ago

web.williams.edu

pvitz• 5 years ago

The excellent book on advanced calculus by Edwin Bidwell Wilson also discusses this method.

Interesting! It wasn't immediately clear what exactly this "trick" is that Feynman was talking about. This document implies that the trick is to differentiate the integral according to another variable (in this case, 't'), and then see where that gets you.

Seeing this sort of creative mathematical process in action makes me think that maybe [1] is right, and math is sometimes more art than science.

[1] https://www.youtube.com/watch?v=Ws6qmXDJgwU&feature=emb_titl...

ui-bello• 5 years ago

Just to explore this idea a little more: insofar as math involves inspiration from the natural world and the logical consequences of axioms, I would consider it a "science" (since these are sort of exploratory and discovered consequences of "facts"). Insofar as it involves redefining axioms, looking at them in a new way, or inventing new idealized objects/methods altogether, I would consider it an art.

xxpor• 5 years ago

I saw this question posed a few weeks ago and it broke my brain for a few days pondering it:

Is math invented or discovered?

koheripbal2• 5 years ago

It is discovered. Two alien species could effectively communicate with math alone (after exchanging notation translate keys).

protonfish• 5 years ago

A flat head screwdriver was designed to insert and remove screws. But it can also be used to open a paint can! OMG! Is the flat head screwdriver invented or discovered?

Our mathematical system is an invented human language. We know that all symbolic systems with sufficient complexity are equivalent (see Turing machines.) Finding an arbitrary one to be useful and flexible is not evidence of magic.

tomrod• 5 years ago

Eh,this is just due to linguistic ambiguity. The syntax of Mathematics (as a language) is used to describe relationships and properties that are discovered.

joppy• 5 years ago

I think both! There are parts of mathematics that just look like truths that have been waiting the whole time to be discovered. On the other hand, people invent problem solving techniques which definitely feel more like inventions than discoveries. Then in the middle, there are made-up mathematical structures introduced to bridge between two “clearly discovered” canonical objects, but this made-up structure certainly has the invented flavour.

So I think it is a continuum, and really fantastic mathematics will feature ideas from all the way along the spectrum: “discoveries” for the beauty, “inventions” for the problem solving, and the “in between” for the subtlety and art.

chronolitus2• 5 years ago

Are maps invented or discovered?

falcrist• 5 years ago

It's almost like you're backing up a bit and taking a different route forward to see if it gets you around a roadblock.

hpcjoe2• 5 years ago

I used this "trick" in many contexts in grad school. Later on, I learned that my bible, Gradsteyn and Rhyzik, also used similar techniques for some of the integrals. I don't have the reference for this, it was verbally conveyed by a professor to me.

I used this in my thesis, in comparing an analytical solution to a problem to a numerical solution, in order to determine some parameters of the numerical solution for idealized wavefunctions. My simulations needed non-idealized wavefunctions, and this mechanism enabled me to optimize parameters for this, and set approximate error bounds.

It (math) really is a science, but there is a strong aspect of artistry involved.

tobmlt• 5 years ago

Oh man, Gradsteyn and Rhyzik!

My masters was about modifying potential flow singularities (Singularities to cancel other singularities... eh hem, I was young) to model vortices shed from blunt surfaces - part of fast/cheap performance prediction for wave energy converters. Didn’t work amazingly well physically, but I will never forget the fun I had that summer figuring out some how to work with those singular integral equations. Working on a set of terms until at long last a form emerged that matched with R&G was such a breakthrough moment!

dynamic_sausage• 5 years ago

These are standard for a (graduate) course of real analysis — see for instance section 2.3 and exercises after it in Folland, "Real analysis".

The reason they are not usually covered in calculus is, to justify such differentiation, one needs the notions of Lebesgue integral and measure. The Riemann integral from calculus courses is just not robust enough. Of course, if the function inside the integral is nice enough, nothing bad happens, and the differentiation is valid.

r-zip• 5 years ago

Under certain technical conditions, differentiation under the integral sign also works for Riemann integration (see Marsden & Hoffman). There's no need to develop Lebesgue theory to demonstrate this technique in a calculus course, but uniform convergence must be understood.

judofyr2• 5 years ago

Alternatively you can study physics and then you don't need to worry about these tiny details.

I took a course on "mathematical methods in physics" which covered some complex analysis, and my math friends where shocked at how non-rigorous we were going through the theorems. Luckily for physicists these techniques tends to be valid because functions from the real world are well-behaved. For me personally it was so fun with a course where we did advanced mathematics for "practical" problems.

anon_tor_123453• 5 years ago

one needs the notions of Lebesgue integral and measure. The Riemann integral from calculus courses is just not robust enough.

definitely not the case. leibniz's rule

https://en.wikipedia.org/wiki/Leibniz_integral_rule#Proof_of...

only requires fubini's theorem for exchanging order of integration

https://en.wikipedia.org/wiki/Fubini%27s_theorem#Riemann_int...

which i'm pretty sure everyone learns in multivariable calc.

i personally learned it from apostol's calc (not analysis) books.

Bahamut• 5 years ago

This is not true - it's not necessarily obvious that things should hold when you're dealing with infinity as one of the bounds. In fact, there are cases where interchanging the order is not allowed when you're in that situation depending on the integrand.

marcodiego• 5 years ago

I remember seeing someone use this trick to integrate an expression that integrals.wolfram.com couldn't.

Anyone knows if is there any symbolic math package that implements this 'trick'?

judofyr2• 5 years ago

Very neat trick! My first reaction was that this must be related to to the Laplace transform, and looking through the Wikipedia article seems that this is basically the same trick: https://en.wikipedia.org/wiki/Laplace_transform#Evaluating_i...:

    \int f(x) g(x) \dx = \int L[f] L^-1[g] \dt

In their sine wave example we have f(x)=sin(x) and g(x)=1/x, and luckily for us the inverse Laplace of 1/x is just 1 (at least from 0 to \infty). I've learnt that finding the inverse Laplace is practically impossible (no good algorithm), but the regular Laplace can be often be found by "just" integrating. So I'm guessing this technique is mostly useful when we have a term with 1/x^n since the inverse is trivial.

I had no idea Laplace transforms could do this so this was a nice discovery!

kkylin• 5 years ago

That's indeed a neat property!

Re: "no good algorithm for inverse Laplace" -- there are certainly reasonably good numerical algorithms based on evaluating contour integrals (see, e.g., https://en.wikipedia.org/wiki/Inverse_Laplace_transform). The inversion formulas are not usually taught in undergrad courses anymore (at least not in the US) because complex function theory has been largely taken out of undergrad engineering curricula, and even for math majors is very much optional.

andi9993• 5 years ago

So are there any good example where this trick gives you answers to otherwise very hard problems?

qubex2• 5 years ago

The academic oeuvre of Dr Feynman (Nobel laureate) himself, largely.

andi9993• 5 years ago

His academic work is not based on/essentially a clever integral trick.

adyavanapalli• 5 years ago

I don't know if this counts, but I solved [problem A5 on the 2005 Putnam exam](https://kskedlaya.org/putnam-archive/2005.pdf) using this method.

andi9993• 5 years ago

So what parameter did you introduce?

emilfihlman• 5 years ago

Is there an error on the second page where it says:

"This clearly converges for all t>=0, and our aim is to evaluate G(0)."

It clearly converges for all t>0, and it would be reasonable to do limit analysis, but I don't quite see how we could say it "clearly converges" for >=.

spekcular• 5 years ago

Break the integral from 0 to infinity as a sum over the intervals demarcated by zeros of (sin(x))/x. Since the integrand is always positive or always negative on such an integral, and the area under the curve is decreasing in absolute value, you can use the alternating series test. (This is a bit clearer if you graph it with e.g. Wolfram Alpha.)

Note that it does not converge absolutely, however.

kgwgk• 5 years ago

“The story is told of G. H. Hardy (and of other people) that during a lecture he said ‘It is obvious… Is it obvious?’ left the room, and returned fifteen minutes later, saying ‘Yes, it's obvious.’ I was present once when Rogosinski asked Hardy whether the story were true. Hardy would admit only that he might have said ‘It is obvious… Is it obvious?’ (brief pause) ‘Yes, it's obvious.’”

fooofw• 5 years ago

This trick can also be used with the Gaussian integral to compute similar integrals of a Gaussian multiplied with a power of the integration variable [1], e.g. the integral of x*exp(-x^2).

[1] https://en.wikipedia.org/wiki/Gaussian_integral#Integrals_of...

hpcjoe2• 5 years ago

Yeah, I used this example in classes I taught. Very awesome application of this technique.

mjcohen• 5 years ago

Of course the indefinite integral of x*exp(-x^2) is elementary.

xdavidliu• 5 years ago

I think parent meant to say _even_ powers like x^(2n) exp(-x^2), which can be done by differentiating exp(-Bx^2) and evaluating at B = 1.

BeetleB3• 5 years ago

While I'm not sure I've seen it in a mathematics textbook, they use this trick all the time in physics. And I'm pretty sure I've had some engineering professors show it in class as well.

goerz2• 5 years ago

Can someone explain to me how Int[0,∞](t^{n+1} x^n e^{-tx} dx) = Int[0,∞](x^n e^{-x} dx)? That is, the last two equations of section 1.

anon_tor_123453• 5 years ago

Let t -> 1 which you can do because this entire trick is about interchanging limits (since both an integral and a derivative are limits)

goerz2• 5 years ago

Ah, thanks! So it's just "has to hold for all values of t > 0, including for t = 1"

raldi• 5 years ago

The second equation of Section 1 says one can "easily" see how

    ∫0->∞ of x*e^-tx dx

equals -1/t², but I'm just not seeing it. Can somebody help me out here?

mvanaltvorst• 5 years ago

You differentiate both sides of the equation given above with respect to t. Differentiating 1/t yields -1/t^2. On the LHS you apply the differentiation inside the integral.

greens• 5 years ago

∫0->∞ x*e^-tx dx = -∂_t ∫0->∞ e^-tx dx = -∂_t (1/t)∫0->∞ e^-u du = -∂_t (1/t) = 1/t^2

MaxBarraclough• 5 years ago

See also the Feynman's Integral Trick thread from 8 days ago:

https://news.ycombinator.com/item?id=26040353

selfishgene2• 5 years ago

Best place to learn about a lot of topics in math including the necessary background for understanding the Feynman trick and related maneuvers is Terrence Tao's website:

https://terrytao.wordpress.com/

Measure theory in particular:

https://terrytao.files.wordpress.com/2011/01/measure-book1.p...

quasirandom• 5 years ago

I think part of the reason Feynman got as far as he did--apart from his unusual innate talents--was his skepticism of formality.

Measure theory will tell you exactly when this result is true, but it is possible to grok the result with only a basic understanding of differentiation and integration. Feynman called this "the Babylonian approach" to mathematics.

    F(t) = integral(a,b) f(t, x) dx
         ~ sum(i) f(t, xi) * Dx

    F'(t) ~ (F(t+Dt) - F(t)) / Dt
          = integral(a,b) f(t+Dt, x)-f(t, x) dx/Dt
          ~ sum(i) (f(t+Dt, xi) - f(t, xi))/Dt Dx
          ~ sum(i) f'(t, xi) Dx
          ~ integral(a, b) f'(t, x) dx

concreteblock• 5 years ago

Feynman was brilliant but I don't think his skepticism of formailty sets him aparts from other good mathematicians or even a typical good mathematics student.

https://terrytao.wordpress.com/career-advice/theres-more-to-...

amirkdv• 5 years ago

Terrence Tao's measure theory text is one of the best undergraduate math textbooks I've ever read. It teaches you not just about the subject, but about how to approach it the way a mathematician does.

selfishgene2• 5 years ago

That is exactly why I recommended it along with his entire website. But thanks for adding the comment.

Suggested edit:

"... the way a world-class mathematician does."

(because it may have a lot to do with why his book is so much better than other texts on this topic like Rudin, Royden, and Cohn.)

tildedave• 5 years ago

Kevin Conrad has a more in-depth paper on this at roughly the same maturity level, also motivated by Feynman's story: https://kconrad.math.uconn.edu/blurbs/analysis/diffunderint....

Differentiation Under Integral Sign (2015) [pdf]

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