Summation series ($\Sigma$) is to Integral ($\int$)... as Product series ($\Pi$) is to ??

Question

If a Summation series ($\Sigma$) is to an Integral ($\int$)... is there a corresponding concept for a Product series ($\Pi$)?

Summation series ($\Sigma$) is to Integral ($\int$)... as Product series ($\Pi$) is to ??

This would be multiplying all the points of a function together to arrive at a result. If there were any points where the function was zero, then the equation would equal zero. The idea being to follow along the curve similar to an Integral.

This question is related to another question I am asking regarding a plane wave intersecting with a curve: Intersection of plane wave surface and a curve

Answer 1

If you want, you can write a product $\prod a_i$ with positive terms $a_i>0$ as $$e^{\sum \log(a_i)} .$$

Now by your own analogy, this is just the exponentiation of an integral of the $\log$.

Answer 2

$$\log\left(\prod f(x)\right)=\sum\log f(x) \implies \prod f(x)=\exp\left(\sum f(x) \right)$$ Now we see that, in some sense, $\prod \to \exp(\int)$

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This is formalized by the so-called "Product Integral" where
$$\prod_a^b f(x) ^{dx} =\lim_{\Delta x \to 0} \prod f(x_i)^{\Delta x} = \exp\left(\int_a^b \log f(x)\right) dx$$ So nothing fundamentally new is needed

Answer 3

You can get from a product to a sum via logarithms: $$ \log\left(\prod_{n=1}^{N} a_n\right) = \sum_{n=1}^{N} \log(a_n). $$ An infinite product is said to converge if and only if the corresponding sum converges, in which case $$ \prod_{n=1}^{\infty} a_n = \exp\left( \sum_{n=1}^{\infty} \log(a_n) \right).$$ Since a sequence $a_n$ is "just" a function from the natural numbers, i.e. $a_n = a(n)$, we could replace the natural numbers with an arbitrary set $X$, and replace $a_n = a(n)$ with a function $f(x)$. If you are really careful about convergence, you might end up with something like $$ \prod_{x\in X} f(x) = \exp\left( \int_{X} \log(f(x) \right).$$ Since the product reduces to some kind of integral, I am not sure that you need a new notation or definition, thus the analogy $\sum:\int :: \prod:?$ doesn't seem to require a definite answer.