Date: 2016-12-16 14:08 From: James Propp <jamespropp@gmail.com> Relatedly: Why is it so easy to obtain a formula for f(n)-f(n-1) given a formula for f(n), but so hard to obtain a formula for f(1)+f(2)+...+f(n) given a formula for f(n)? :-) Jim Propp https://www.ncbi.nlm.nih.gov/pmc/articles/PMC411178/pdf/pnas00013-0045.pdf :-) On Friday, December 16, 2016, Dan Asimov <asimov@msri.org> wrote:

Why is it usually so easy to differentiate a function defined by an exact formula, but so much more difficult to integrate?

If this question can be made rigorous, how might that be done?

(And if so, what is the rigorous answer, or at least a method of approaching it?)

—Dan

A sort of counterexample is that if you can differentiate d/ds HurwitzZeta(s,a), you can do fancy integrals of t^s log t! . http://gosper.org/filds.pdf pp 1, 5. And sure enough, that derived function is hard to compute. --rwg