A common example of integration by parts used in many Calculus II classes has students compute
by integrating by parts twice, then rearranging terms to arrive at a solution. This technique is handy for many functions whose derivatives eventually repeat, that is, functions satisfying
for some integer n and some constant c. (Question: Is there a name for such functions? I feel like I should know this.)
Convolution is, among other things, a very useful way of computing inverse Laplace transforms, and appears in many other applications in functional analysis. One of the first exercises asked for the following:
If , what is ?
This requires one to compute The “best” way to handle this is to use the difference formula for the sine function and rewrite the integral as
and integrate each term (either by substitution or by using an identity, then substitution). Another approach, which several of my students took, is to try integrating by parts twice, as in the first example above. Unfortunately, the technique fails in this case. (Try it yourself and see!) One student came to me wondering why. My answer: I don’t know.
After discussing the problem for a while, we decided (i.e., I decided) this would be a great problem for her to explore. I think the more general question about integrating products of these functions by parts could be an interesting exploration for both of us. I haven’t had time to play around with the problem since our initial discussion, but I suspect there’s something deeper lurking in the background. As soon as we find out, I’ll post a follow-up.
[This problem was suggested by Jolie Roat, and I believe her work will become a presentation in the spring, so don't go spoiling all the fun by posting a complete solution!]