Talk:Product (mathematics): Difference between revisions
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Keilandreas (talk | contribs) |
Keilandreas (talk | contribs) removed my own former comment since it could be resolved by linking to product integral |
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== Continuous Product / Multiplical == |
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What about adding the "Continuous Product" (aka "Multiplical") to this article? It is the continuous equivalent to the product sequence (or the multiplicative equivalent to the integral) and very useful esp. when working with probabilities. See [http://dx.doi.org/10.1007/BF02365376] for more detailed information (exact definition, formulas, examplary usage). |
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I will give a preliminary definition here and suggest to work on it and include it in section 3 of this article and also in [[Multiplication]]: |
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Let <math>f \colon \mathbf{R} \to \mathbf{R}</math> fulfill certain boundedness and positivity conditions on an interval [a,b]. |
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The continuous product (a.k.a. "mulitplical") can then be defined in a Riemannian sense as |
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<math> |
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\prod_a^b f(x) {}^{dx} = \lim_{n \rightarrow \infty} \prod_{i=1}^n f \bigl( a + i \cdot \Delta x(n) \bigr)^{\Delta x(n)} \quad\text{with}\quad \Delta x(n) := \frac{b-a}{n} |
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</math> |
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if the limit exists. |
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An alternative definition is derived from the discrete equivalence <math>\prod_i f_i = \exp ( \sum_i \ln f_i )</math> and is given by |
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<math> |
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\prod_a^b f(x) {}^{dx} = \exp \left( \int_a^b \ln f(x) dx \right) . |
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</math> |
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The definition of the continuous product is the continuous equivalent of the indexed product operator and the "product-wise" equivalent to the integration: |
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<math> |
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\begin{array}{c|cc} |
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& \text{additive} & \text{multiplicative} \\ \hline |
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\text{discrete} & \sum_{i=a}^b f(i) & \prod_{i=a}^b f(i) \\ |
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\text{continuous} & \int_a^b f(x) dx & \prod_a^b f(x) {}^{dx} |
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\end{array} |
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</math> |
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The most common notations for the continuous product sign seem to be the Pi-like symbol also used for product sequences and <math>\mathcal{P}</math>. I have chosen to use the former one because it is less disturbing for readers not familiar with the concept, because it's more intuitive, and because <math>\mathcal{P}</math> is already used for other things (like the power set or individual variables of mathematical texts). In the long run, a distinct, new symbol for the continuous product would be desirable (just like the integral sign for the continuous summation). |
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[[User:Keilandreas|Keilandreas]] ([[User talk:Keilandreas|talk]]) 10:54, 26 March 2010 (UTC) |
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