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What are your favorite applications of integration by parts?

(The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!)

Thanks for your contributions, in advance!

  • 49
    It can also be a good career move. A (likely apocryphal) story goes: when Peter Lax was awarded the National Medal of Science, the other recipients (presumably non-mathematicians) asked him what he did to deserve the Medal. Lax responded: "I integrated by parts."2011-04-24
  • 3
    Great story, Willy.2011-04-28
  • 20
    Two more stories: 1. Supposedly when Laurent Schwartz received the Fields Medal (for his work on distributions, of course), someone present remarked, "So now they're giving the Fields Medal for integration by parts." 2. I believe I remember reading -- but have no idea where -- that someone once said that a really good analyst can do marvelous things using only the Cauchy-Schwarz inequality and integration by parts. I do think there's some truth to that.2011-10-11
  • 4
    More physics, but it's useful in the derivation of the [Euler-Lagrange equation](http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation#Statement), which itself is very nice.2013-01-12
  • 3
    @WillieWong Your comment is quoted in the book "Physics from Symmetry" https://books.google.de/books?id=_vLLCQAAQBAJ&pg=PA256&lpg=PA256&dq=A+%28likely+apocryphal%29+story+goes:+when+Peter+Lax+was+awarded+the&source=bl&ots=TbPF0vhSni&sig=tnQ4yvjLZ2yso8ch_TnJ9NayylY&hl=de&sa=X&ei=QE2FVeyLAsWWsAHgwIXIBw&ved=0CCoQ6AEwAQ#v=onepage&q=A%20%28likely%20apocryphal%29%20story%20goes%3A%20when%20Peter%20Lax%20was%20awarded%20the&f=false2015-06-20
  • 0
    @WillieWong: I can't understand your comment clearly. What's the good career move ?2016-04-05
  • 0
    @ArkaKarmakar: It == "Integrating by parts".2016-04-08
  • 0
    @WillieWong: I am stupid than most of the users here, I still can't understand how integrating by part helped Peter Lax.2016-04-08
  • 1
    @ArkaKarmarkar: Peter Lax intimated that the work he did that led to his being awarded the National Medal of Science was essentially integration by parts. If you win a National Medal of Science then this helps your career as a mathematician.2016-04-08

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