One definition of integration over a continuous interval [a,b] into n subintervals with equal width $\Delta x$, and from each interval choose a point $x_i^*$. Then the definite integral of $f(x)$ from a to b is $$\int_{b}^{b}f(x)dx = \lim_{n\to\infty}\sum f(x_i^*) \Delta x$$.
What happens if the summation ($\sum$) is replaced with a product ($\prod$)? Is there a name for this type of infinite product?
For positive quantities, $\prod_i x_i=\exp\sum_i\ln x_i$ allows us to make a "continuous product" by exponentiating an integral. If quantities are real but allowed to be negative, we run into a problem: can we count the number of sign changes, when it might be countably or uncountably infinite? But with complex numbers we can write $\prod_i r_i\exp\mathrm{i}\theta_i=\exp\sum_i(\ln r_i+\mathrm{i}\theta_i)$, which again allows the exponentiated-integral trick to work. It comes up a lot in quantum field theory.
