Can there be a symbol for continuous product?

Asked Jul 10 '17 at 06:05

Active Jul 10 '17 at 06:31

Viewed 1,778 times

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We know that continuous version of $\sum$ is $\int$, but, can there be a continuous version of $\Pi$?

edited Jul 10 '17 at 06:31

asked Jul 10 '17 at 06:05

ankit

2,591

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Equally loosely speaking, $,\prod = e^{\sum \ln },$. – dxiv Jul 10 '17 at 06:07
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$e^{\int \log}?$ ${}$ – Chris Jul 10 '17 at 06:07
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Related: What is to geometric mean as integration is to arithmetic mean? – Jul 10 '17 at 06:17
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@dxiv That works if we product integrate a positive (or at least nonnegative) function, but the product integral is more interesting for operators (e.g. matrices) that don't commute. – md2perpe Jul 10 '17 at 07:36

1 Answers1

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There is indeed: it is called the product integral.

answered Jul 10 '17 at 06:09

user1337

24,859

Hah, my off-the-cuff idea actually means something. Also, wow. (+1) – Chris Jul 10 '17 at 06:10
I guess there won't be any for exponentiation or tetration, right? – ankit Jul 10 '17 at 06:27
@ankit I'm not aware of continuous analogs for these. – user1337 Jul 10 '17 at 06:29
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@ankit I don't suppose you can define any continuum-limit if the underlying operation is not associative. – leftaroundabout Jul 10 '17 at 16:09