Reimann sums, okay. How about a Reimann product ?

[pellman]

Reimann sums, okay. How about a "Reimann product"?

An integral is a sort of "continuous sum". Very roughly, the sum Σ_k f(x_k) Δx goes over to the integral ∫f(x)dx when the number of terms becomes infinite while Δx goes to zero.

What about a similar "continuous product"? If we have the product

Π_k [f(x_k)] ^P_k

and let the exponents P_k go to zero while the number of products goes to infinite (and the range of x_k is fixed as in an integral) what sort of an animal do we get?

I realize that we can turn this product into a sum by taking the anti-logarithm. Then the continuous limit of this product is e raised to an integral with ln(f) in the integrand. However, I'm curious if the properties of these sorts of entities are known without resorting to integrals. And is there a notation for them?

 

[Ambitwistor]

I don't think you want to let the exponents go to zero; then each factor in the product will approach 1.

I don't know if it's really what you're looking for, but something a little analogous to a "continuous product" is the exponentiation of an element of a Lie algebra to get an element of a Lie group.

As a simple, scalar example, consider the n-fold product of (1+x/n), which is Π_n (1+x/n)ⁿ. The limit as n goes to infinity is just the function exp(x).

 

[pellman]

Thanks, Ambitwistor.

Having thought it over, I see now that these products have probably not been studied in themselves since they can always be reduced to an integral. Like so, ..

Π_k [f(x_k)] ^P_k
= exp[ ln ( Π_k [f(x_k)] ^P_k ) ]
= exp[ Σ_k P_kf(x_k) ]
= exp[ Σ_k α(x_k)f(x_k) Δx ] , P_k = α(x_k)Δx
--> exp[ ∫ α(x) f(x) dx ]

which shows why I said the exponents go to zero.