I know that the Euler-Maclaurin formula establishes a relationship between sums and integrals, but is there some sort of formula that establishes a relationship between products and integrals? I don't mean the product rule, for products of functions. I mean a way to convert the sort of product you use the Π notation for into an integral. Specifically, a product of all the values of a function, from f(1) to f(n), into an integral.

Sorry if this is a silly question; I'd still like to know the answer. Thanks very much.


1 Answer 1


Assuming $f(i) > 0, \forall i$, use can use the identity $$\prod_{i=1}^n f(i) = \exp \left[\sum_{i=1}^n \log f(i)\right]$$ and then employ Euler-Maclaurin for the sum on the right hand side.

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    $\begingroup$ This applies perfectly to what I'm doing; thank you so much! $\endgroup$
    – Annick
    Aug 14, 2012 at 22:00
  • $\begingroup$ @Annick If your question's answered well, don't forget to click the check mark to mark the answer correct! :) $\endgroup$ Aug 14, 2012 at 22:39
  • $\begingroup$ Sorry I'm not familiar with the notation. What is exp[x]? Does that mean e raised to the power of [x] $\endgroup$ Jun 4, 2016 at 20:15
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    $\begingroup$ @AlbertRenshaw: yes, exp is just another notation of $e$ to the power of ... . The square brackets $[]$ have the same meaning as usual brackets $()$. $\endgroup$
    – Fabian
    Jun 4, 2016 at 22:04
  • $\begingroup$ @Fabian thank you very much! $\endgroup$ Jun 4, 2016 at 22:04

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