# Thread: Infinite analogue to product

1. ## Infinite analogue to product

I don't really know where to put this however I thought that it might fit in here.

We have that an integral is often considered as a sort of continuous analogue to the notion of a summation, however is there any way in which one can achieve a continuous analogue to a product? I suppose in a more pressing sense, would such an analogue have any real meaning, as in could it be sensibly defined (irrespective of whatever it might be used for)?

I was considering the idea that if one already has the generalisation of the sum then one could use logarithms to extend this; is this in any way sensible?

2. Originally Posted by ihateyouall
I don't really know where to put this however I thought that it might fit in here.

We have that an integral is often considered as a sort of continuous analogue to the notion of a summation, however is there any way in which one can achieve a continuous analogue to a product? I suppose in a more pressing sense, would such an analogue have any real meaning, as in could it be sensibly defined (irrespective of whatever it might be used for)?

I was considering the idea that if one already has the generalisation of the sum then one could use logarithms to extend this; is this in any way sensible?
"...is there any way in which one can achieve a continuous analogue to a product?" I think you meant to say sum instead of product.

Your question is sort of broad. I think the answer is yes (not as a formula, but as a series of methods - I'd like to see specific examples on what you're talking about).

I reread your post and I understand it better. There is a concept based on capital pi which correlates to the sigma sum which might be the answer to your question as far as products go.

3. I know about the use of capital pi notation for expressing countable products, if that is what you are referring to.

When one does mechanics it is usual that the first idea of the center of mass is introduced as a weighted average of finitely many points. We have a good understanding of what it means to take finitely many objects and then sum them, we even have a good knowledge of what it means to take countably many objects and sum them (in the sense that we can define infinite sums through limits). However when one gets further in mechanics it becomes clear that one needs to address continuous surfaces, here one uses an integral to 'sum up' a continuous surface; in this sense one has an analogue of the sigma but in a continuous (uncountable) sense. My question is about extending the idea of the capital pi notation in a similar way. Is there some means by which this can be done?

4. you can take a look in product integral from wikipedia
Product integral - Wikipedia, the free encyclopedia

and non newtonian calculus
http://en.wikipedia.org/wiki/Non-Newtonian_calculus