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

summation integration products

Solution 1:

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.

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