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


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    $\begingroup$ Equally loosely speaking, $\,\prod = e^{\sum \ln }\,$. $\endgroup$ – dxiv Jul 10 '17 at 6:07
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    $\begingroup$ $e^{\int \log}?$ ${}$ $\endgroup$ – Chris Jul 10 '17 at 6:07
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    $\begingroup$ Related: What is to geometric mean as integration is to arithmetic mean? $\endgroup$ – Rahul Jul 10 '17 at 6:17
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    $\begingroup$ @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. $\endgroup$ – md2perpe Jul 10 '17 at 7:36

There is indeed: it is called the product integral.

  • $\begingroup$ Hah, my off-the-cuff idea actually means something. Also, wow. (+1) $\endgroup$ – Chris Jul 10 '17 at 6:10
  • $\begingroup$ I guess there won't be any for exponentiation or tetration, right? $\endgroup$ – ankit Jul 10 '17 at 6:27
  • $\begingroup$ @ankit I'm not aware of continuous analogs for these. $\endgroup$ – user1337 Jul 10 '17 at 6:29
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    $\begingroup$ @ankit I don't suppose you can define any continuum-limit if the underlying operation is not associative. $\endgroup$ – leftaroundabout Jul 10 '17 at 16:09

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