11

In the topic of integration and anti-derivatives in Calculus we come across cases where the attempt at integration by parts brings us back to the original integral, the most basic example being $\int e^{a x} \cos(b x) \,dx$. Is there a standard name for this type of integrals? I have been calling them "self-referral", but the only usage of this expression that I can find is in health insurance! Have you seen a name in standard textbooks? I have seen "circular integrals" online. What do you call them? Does anyone call it "self-referential"? (Stewart does not have a name for it. Self-referential is an expression in logic. Circular integral might be used for contour integrals.)

Michael Hardy
  • 1,885
  • 11
  • 24
Maesumi
  • 1,380
  • 9
  • 19
  • 7
    I have never heard of a term for such a thing, nor do I think that they are important enough to bother with a specific name. However, I would avoid the term "circular integral", as I would understand such an integral to either be an integral over a circle, or an integral related to some specific subset of elliptic integrals (not that I know what specific set this would be---that would just be my assumption about the term). – Xander Henderson Jan 19 '24 at 18:33
  • 2
    "self-referential" seems fine. I love how those turn into algebra problems, which is part of what I emphasize. – Sue VanHattum Jan 19 '24 at 19:04
  • 18
    Giving it a name would give the (wrong) impression that it's a feature of the integral itself, whereas it's only a feature of this particular technique we're using to derive a closed form expression for this integral. PS. Justin Skycak's answer provides a name which is free of this issue. – Michał Miśkiewicz Jan 19 '24 at 20:11
  • 1
    Note the example you gave in your question can also be solved by using complex numbers which might be faster than the self-referential integral you speak of. I can't remember if this idea was in Stewart's Calculus text as well. Look at $$e^{x(i+1)}=e^x\cos x +ie^x \sin x$$ and then take the real or imaginary part. Hope this is another tool for your integral solving! – Matthew Albano Jan 19 '24 at 20:46
  • 1
    In my high school calculus class we called it "recovering the integral back". – Steven Gubkin Jan 19 '24 at 21:53
  • 2
    Call me a simpleton, but my programming background tells me that you arrive at an equation involving recursion. – TAR86 Jan 20 '24 at 11:08
  • 10
    On Mathematics Stack Exchange, it was referred to as "Integration by parts with déjà vu". I found this terminology amusing enough that I use it myself whenever I tutor school students. – Joe Jan 21 '24 at 00:23
  • 1
    A colleague at my university calls them "ninja integrals" (the result seems to appear out of nothing), and I have adopted that term. – Torsten Schoeneberg Jan 21 '24 at 16:42
  • 2
    FWIW, he German name for these integrals translates literally to “phoenix from the ashes” or just “phoenix”. That doesn’t seem to be used in English though. – Wrzlprmft Feb 12 '24 at 16:53

2 Answers2

10

To my knowledge, this is most commonly known as "cyclic" integration by parts.

Justin Skycak
  • 9,199
  • 4
  • 27
  • 47
  • 1
    Seems like a relatively new name. I can't find this in any books. And the earliest such use of this name I could find was this 2012-09-11 video: https://www.educreations.com/lesson/view/cyclic-integration-by-parts/1469863/ – user182601 Feb 12 '24 at 00:54
1

From Does integration by parts with "deja vu" have a name?:

  • Sheard ("Trick or Technique?", 2009) calls it the one-step algebra trick;
  • OP says he saw it called integration by parts with "déjà vu";
  • Glen Wheeler suggests calling it absorption.
user182601
  • 171
  • 4