Mike Stay (staym@datawest.net) asks: << sum is to integral as prod is to what? I.e. is there a name for exp(integral(log(f(x)),dx))? Where do I worry about branch cuts in that expression?
These are often called "product integrals" and have application to the branch of probability called survival analysis. (As I gather you have already realized, given a positive function f defined on [a,b],the product integral of f over [a,b] is the limit as n -> oo of sum from 1 to n of f(x_i)^deltax_i, wherex_i and deltax_i are defined as in the Riemann integral, with the mesh of the partition going to 0 as n -> oo.) (Back in the days when I thought I was the first person to invent this idea, I was 16 and it was my Westinghouse project. As I found out when I got to college, Gerry Sussman thought of the very same idea for *his* Westinghouse project the same year. I still don't know who first invented these.) This can be used to calculate the *geometric* mean of a function over an interval; e.g., the GM of x over (0,1) is 1/e. These also work fine in the complex plane as well, as long as log is continuous and well-defined on the domain D of f. One way to ensure this is by picking D to be any simply-connected region of C - {0}. More generally, D can be chosen to be any simply-connected region of the Riemann surface for log. Once the domain has been chosen, there is a countable set of choices for which branch of log you prefer to use; they all differ of course by an integer multiple of 2 pi i. Just for fun, What is the geometric mean of all the complex numbers in the disk 0 < r <= 1, where we define log for -pi < theta < pi so that it's real for positive reals ??? --Dan