Integrate Small Numbers: General Theory & Results

[jostpuur]

Rough idea behind integration is to sum lot's of small numbers (close to zero). Some problems lead to situations where you have to multiply lot's of numbers close to one. Is there any general theory of such products? Important results or tools?

 

[Eighty]

The first is called a series or an infinite sum. The second is called an infinite product.

 

[HallsofIvy]

I doubt that you can find a great body of work on the subject since taking a logarithm reduces to the sum and thus a standard integral.

 

[jostpuur]

In fact the main motivation for this product question comes from quantum physics,
$$\prod_{k=0}^n \exp(iH_k \Delta t /\hbar)$$, where product cannot be turned into sum in the exponent always since $$H_k$$ do not always commute.

So more generally, the product could be a product of matrices. Of course, if there is some explicit problem, you can see how things go and try to do something tricky, but I just asked this in a hope that there could to be some general results.

The infinite products only weren't precisly what I was after, but the Wolfram site did look interesting anyway.