I was reading one of my books by Feynman (I think it was Lectures in Computation) and he mentioned that there existed an “effective procedure” for determining the integral of a function. Now in school, it was a common assignment to determine the approximate value for a definite integral and that is a simple problem, but it appeared that Prof. Feynman was talking about indefinite integrals, and that by “effective procedure”, he meant algorithm. Do you know what he was talking about and if so, what is this effective procedure?
Yes, it’s in Lectures in Computation; specifically, he says “Specifically, any integral which can be expressed in terms of a pre-defined list of elementary functions – sines, exponentials, error functions and so forth – can be evaluated by an effective procedure… We have to thank a guy named Risch for this.” He is referring to the Risch algorithm (clearly), about which there is more information on Wikipedia.
While the Risch algorithm is a very nice theoretical result, it’s pretty rare that you need the exact functional form of an antiderivative for any complicated function. Pure mathematicians are generally satisfied knowing whether antiderivative exists (it does for any continuous function) and being able to say something about its properties, and the rest of us are generally more interested in the numeric values of integrals, which can be computed without doing the integration.
edit: One very nice example of the lack of need for an exact functional form is the Gamma function. There is no closed form expression for it, but it’s easy to compute to a very high accuracy and there’s a very well-developed theory about it. Feed that into the Risch algorithm and it’ll just spit out no.
While it is always possible to numerically integrate any function (or at least any function anyone doing numerical integration would ever encounter), an analytic solution (if it exists) is almost always preferable. You can learn a lot more from the analytic form, and it’s also usually easier to calculate (which can be a great boon if you need to do it repeatedly).
Sometimes the best analytic solution available is in terms of things like the Gamma function, but even then, I’d rather have the solution in terms of the Gamma function than just the numbers spit out by a numerical integrator. There is, as you say, well-developed theory around the Gamma function (how it relates to other functions, how it behaves in various asymptotic limits, etc.), and really, when you get right down to it, all the trig functions are is named solutions to otherwise-insoluble differential equations, too, and nobody has a problem with that.
Right on. “Closed form” is always a relative notion; it just means “Can be expressed in terms of things I like”. Everything’s closed form if you like the right things, and isn’t if you don’t.
Of course, so far as the Risch algorithm in particular goes, there’s a particular attached notion of closed form (one which includes the trig functions and not the Gamma function). But there’s nothing absolute about this; one could consider other algorithms to handle other (broader, narrower, or just plain different) classes of functions.