Since a rational functions is analytic, its integral in "the complex domain" has the same expression as in the real domain. So just use partial fractions for every ratio of polynomials, and if the integrand was real, the integral is equal to a real-valued function of real arguments in any case, even if its metamorphosis included a complex stage. (Like the casus irreducibilis of the cubic.) —Dan
On Dec 16, 2016, at 2:52 PM, Henry Baker <hbaker1@pipeline.com> wrote:
I was hinting at the fact that integrate(A(x)/B(x),x), where A(x), B(x) are polynomials in x, is quite easy when B(x) factors into linear factors; use partial fraction expansion. A lot of trig functions crop up when you are unable or unwilling to move into the complex domain and B(x) has quadratic factors.
At 02:27 PM 12/16/2016, Fred Lunnon wrote:
Numerically speaking, integration is in contrast easier than differentiation ... WFL
On 12/16/16, Henry Baker <hbaker1@pipeline.com> wrote:
2. Some closed form integrals may require polynomial root-finding (or root expressing), so you may not like the look of the resulting expression. E.g., you might find floating point approximations to certain numbers instead of a "perfectly precise" answer.
On Friday, December 16, 2016, Dan Asimov <asimov@msri.org <mailto: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?)