I just learned that when a function does not have compact support, you might need to introduce something called "boundary terms" when integrating it by parts. I've scoured the internet trying to learn about them, and I can't find anything about them that doesn't assume that you already know what they are!

Could somebody please explain to me (in a very basic way) 1) what boundary terms are; 2) how a function not having support necessitates them. If this is too involved, can you suggest a book or website where I could learn about them?

  • $\begingroup$ These come from Stokes theorem. This is covered in Michael Spivak's Calculus on Manifolds. $\endgroup$ – ncmathsadist Feb 1 '14 at 1:42
  • $\begingroup$ Recall the definition of improper integral when considering $\int_{-\infty}^{+\infty} u dv$. $\endgroup$ – Hurkyl Feb 1 '14 at 2:25

Integration by parts follows, in a simple sense, from the product rule of differentiation:


Performing an integral (definite, in our case) on both sides of the equation and using linearity to switch the sides of one of the terms, we get:

$$\int_a^b u'v = \int_a^b v'u - \int_a^b (uv)'$$

According to the fundamental theorem of calculus, the last term is just the difference $(uv)(b)-(uv)(a)$, stated now as:

$$\int_a^b u'v = \int_a^b v'u - \left.uv\right|_a^b$$

The last term is what is called, if I'm not mistaken, the boundary term. It's more generally an integral of the product function evaluated on the boundary of the region of the original integral, hence its name.

I actually found your question looking to see if the term I knew for it in Hebrew is the same as in English, and it seems that it is.

If you take the limit of $a,b\to\infty$, for any function with compact support, that term vanishes. For any function without compact support, that term may be non-zero.

Hope this helps.


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