# If integration is a continuous analog of summation (Addition), what is the continuous analog of multiplication (Product)?

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?

• Stieltjes integral? Commented Feb 15, 2019 at 17:13
• Nice question. There is such a thing, and it is called the product integral. Commented Feb 15, 2019 at 18:43
• Yes, this is exactly what I was imagining Commented Feb 15, 2019 at 20:01

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.