="Mike Stay" <staym@datawest.net> 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?
="Richard Schroeppel" <rcs@CS.Arizona.EDU> [editorial comment: Gosper calls these Prodigals. I don't think they have any non-obvious proerties, but it is strange there's no "official" name. --Rich]
I must confess to coining "prodigal" for the "continuous product" (add 'em to get a "prodigal sum"<;-) You can worry about the branch cuts in that expression if you want, but usually I simply offer a prayer to Euler and blithely proceed, then go back later and check results. This may actually be unreasonably effective, because you can define the prodigal directly as a limit, analogous with the integral, instead of as a transform of an integral. (Martian mathematicians started with the prodigal, and then defined the integral by analogy!). While in some sense prodigals are "trivial", I advocate greater use of them because in certain situations they make things clearer and/or suggest interesting ideas that otherwise get lost in complicated expressions. Continuously compounded interest is naturally expressed as a prodigal, as are certain kinds of cumulative probabilities (consider a particle "deciding" whether to decay at each uncountable instant), for examples. Stirling's factorial approximation is basically the Bernoulli expansion of a prodigal. Note that instead of an arbitrary constant offset prodigals have an arbitrary constant scale factor (any application to big-O notation?). Speaking of notation, it's fun to write a "stretchy-P" for the prodigal, analogous to the "stretchy-S" of the integral: /\ |_/ qx | u | \/ (Easy to scribble, but how the heck do you get this graphic into documents?) Note that the "quintessential" qx, the analog of the differential dx, goes in the exponent. The inverse prodigal is of course 1/qx d(ln u)/dx du/(u dx) qu = e = e and there are naturally definite prodigals and the like. A provocative application is extending the idea of defining polynomials as the product of a countable number of zero monomials to defining functions as prodigals with uncountable numbers of "roots". (Note that, unlike polynomials, the function value need not be zero at any "root" point!) For example, x 1 qz x P (x-z) = ---------- 0 x-1 e (x-1) is the function defined by the unit interval "root segment". More generally, a function f(x) can be constructed as a prodigal over ALL "zeros" z (in the complex plane, say) weighted with multiplicity mu(z): mu(z) qz f(x) = P (x-z) Then we can view f and mu as "zero transforms" of each other (with suggestive analogy to Fourier convolution?). But that's about as far as I've managed to plod... maybe someone else will have fun prodding this further. "Enjoy"!