[HN Gopher] Differentiation Under Integral Sign (2015) [pdf]
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Differentiation Under Integral Sign (2015) [pdf]
Author : dualvectorfoil
Score : 67 points
Date : 2021-02-13 14:26 UTC (8 hours ago)
(HTM) web link (web.williams.edu)
(TXT) w3m dump (web.williams.edu)
| andi999 wrote:
| So are there any good example where this trick gives you answers
| to otherwise very hard problems?
| qubex wrote:
| The academic _oeuvre_ of Dr Feynman (Nobel laureate) himself,
| largely.
| adyavanapalli wrote:
| 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.
| dynamic_sausage wrote:
| 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.
| anon_tor_12345 wrote:
| >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.
| r-zip wrote:
| 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.
| judofyr wrote:
| 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.
| sdenton4 wrote:
| It's all fun and games until you're trying to calculate
| trajectories over a Cantor set.
| qubex wrote:
| Or integrate over an interval of surreals.
| BeetleB wrote:
| As someone who loved both math and physics, this was why I
| always found math a bit easier. Everything rests on a solid
| foundation and you can justify each step. When I got into
| higher physics, it was so riddled with intuitive arguments
| as opposed to rigor that I didn't fare so well. I'm sure
| one can find mathematical justifications for their methods,
| but it's not part of the curriculum, and almost none of the
| professors (in a top 10 physics school) knew them either.
| TheOtherHobbes wrote:
| Apart from the occasional Einstein and Newton every
| couple of centuries, physics seems to advance by throwing
| a semi-random selection of PhD dissertations at the real
| world and seeing if any of them happen to match
| experiment.
| BeetleB wrote:
| Case in point: Newton's work was not that rigorous. It
| was not till the 1800's that calculus was put on a firm
| foundation. Of course, things were all different back
| then.
| fooofw wrote:
| 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...
| mjcohen wrote:
| Of course the indefinite integral of x*exp(-x^2) is elementary.
| hpcjoe wrote:
| Yeah, I used this example in classes I taught. Very awesome
| application of this technique.
| emilfihlman wrote:
| 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 >=.
| marcodiego wrote:
| 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'?
| goerz wrote:
| 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_12345 wrote:
| Let t -> 1 which you can do because this entire trick is about
| interchanging limits (since both an integral and a derivative
| are limits)
| goerz wrote:
| Ah, thanks! So it's just "has to hold for all values of t >
| 0, including for t = 1"
| raldi wrote:
| 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?
| greens wrote:
| [?]0->[?] x*e^-tx dx = -[?]_t [?]0->[?] e^-tx dx = -[?]_t
| (1/t)[?]0->[?] e^-u du = -[?]_t (1/t) = 1/t^2
| mvanaltvorst wrote:
| 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.
| pvitz wrote:
| The excellent book on advanced calculus by Edwin Bidwell Wilson
| also discusses this method.
| BeetleB wrote:
| 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.
| judofyr wrote:
| 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 wrote:
| 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.
| MaxBarraclough wrote:
| See also the _Feynman 's Integral Trick_ thread from 8 days ago:
|
| https://news.ycombinator.com/item?id=26040353
| chronolitus wrote:
| 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 wrote:
| 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.
| hpcjoe wrote:
| 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 wrote:
| 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!
| xxpor wrote:
| 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?
| koheripbal wrote:
| It is discovered. Two alien species could effectively
| communicate with math alone (after exchanging notation
| translate keys).
| tsimionescu wrote:
| This is an interesting thought experiment. The space of all
| consistent and potentially useful mathematical constructs
| is gigantic, so I think there would be a good chance that
| two alien species would share almost no mathematical
| constructs, and would require decades or hundreds of years
| to discover - so in this sense, there is a large element of
| invention to mathematics as a human endeavor.
|
| Even for physics, there are often many mathematical
| theories that can be used to model the same physical
| observations (talking about equivalent structures, not
| about competing theories). For example, many problems can
| be described equivalently using vectors, complex numbers,
| or linear algebra. There is a good chance that there are
| many (perhaps infinitely many) other systems that we
| haven't thought about that could be used equivalently.
|
| So, while I agree that ultimately the structures in
| mathematics exist independent of our use of them, so we are
| only discovering pre-existing structures, I would also say
| that new mathematical theories are developed using a
| process that is more similar to invention than to discovery
| (i.e. you can't explore the space of mathematic theories to
| discover new ones, as it is infinite in every direction -
| you can only explore the properties of a structure you
| essentially invent for yourself).
| mxwsn wrote:
| I agree with your logic but not your premise. Two alien
| species could effectively communicate if they happened to
| agree on a shared set of fundamental axioms. The axiom of
| choice is somewhat contentious here on earth, since it
| underlies the Banach-Tarski paradox, and it's not clear at
| all that a sophisticated alien society would have ever
| accepted the axiom of choice into their mathematics.
| joppy wrote:
| 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.
| chronolitus wrote:
| Are maps invented or discovered?
| protonfish wrote:
| 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.
| carapace wrote:
| That thing to which all symbolic systems with sufficient
| complexity are equivalent, discovered or invented?
|
| (Yours is a "turtles all the way down" argument, I think.)
| tomrod wrote:
| 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.
| falcrist wrote:
| 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.
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