Integrating a product of infinitely many terms

[benorin]

Ok, so I want to integrate a general function defined by an infinite product, and let us assume that the product is nice (e.g., absolutely convergent, ect.).

So, without expanding into an infinite sum, how do I evaluate $$\int_{z=a}^{b}\left(\prod_{n=0}^{\infty}(1+f_{n}(z))\right) dz$$

Let z be real or complex, according to your preference.

Thanx, I know you guys will me help out.

 

[Tide]

I think you should work on finding closed forms for the product first.

 

[benorin]

I think you should work on finding closed forms for the product first.

The motivation for the prompt was to find a way to perform said integration when $$\{f_{n}(x)\}$$ is an unknown sequence of functions. Hence finding a closed form for the product is, well, rather difficult.

 

[Tide]

In that case it would seem that your question is akin to asking what is the general result of integrating an unknown function g(x). That's simply not possible except in very special cases where, e.g. you know the result of the integration and are trying to determine the function (e.g. inverse scattering problems, Volterra integral equation etc.)

 

[benorin]

I'm looking for a theorem

I'm looking for a theorem like unto $$\int_{z=a}^{b}\left(\sum_{n=0}^{\infty}f_{n}(z)\right) dz=\sum_{n=0}^{\infty}\left(\int_{z=a}^{b}f_{n}(z)dz\right) \Leftrightarrow \sum_{n=0}^{N}f_{n}(z) \rightarrow F(z) \mbox{ uniformly as } N\rightarrow\infty$$.
Clearly, it is generally true that the intergral of a product is NOT the product of the integrals. But I had hoped for some condition, such as uniform convrgence, and something else, such as the interchange of the order of summation and integration, that would make the integration of infinite products nice.

 

[Tide]

I don't see how to do it. Anyone else?

 

[Jonny_trigonometry]

Since you're going to infinity for the sum, could you change it to an integral?