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infinite_product_of_complex_numbers [2015/04/15 17:09] nikolaj |
infinite_product_of_complex_numbers [2016/04/02 14:53] nikolaj |
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| === Discussion === | === Discussion === | ||
| - | >todo: Interestingly, I think I see the nLab doesn't want to allow e.g. $\prod_{n=1}^\infty (17-n)$ to be zero. | + | >todo: Interestingly, I think I see the nLab doesn't want to allow e.g. $\prod_{n=1}^\infty (17-n)$ to be zero. (Reference below) |
| - | + | ||
| - | http://ncatlab.org/nlab/show/infinite+product | + | |
| I recon infinite products may arise when x is written as $111\cdots11x$ and 1 is represented as something. | I recon infinite products may arise when x is written as $111\cdots11x$ and 1 is represented as something. | ||
| Line 16: | Line 14: | ||
| >can Euler products arise in this way? | >can Euler products arise in this way? | ||
| - | And they may arise as determinants infinite dimensional operators, i.e. products of eigenvalues. | + | And they may arise ans determinants of infinite dimensional operators, i.e. products of eigenvalues. |
| === Theorems === | === Theorems === | ||
| Line 30: | Line 28: | ||
| Also, from the sum formula with $a_n=\prod_{k=K}^{n-1}b_k$, we get | Also, from the sum formula with $a_n=\prod_{k=K}^{n-1}b_k$, we get | ||
| - | ^ $\prod_{k=K}^\infty b_k=\lim_{n\to\infty}a_n=\prod_{k=K}^{M-1}b_k+\sum_{n=M}^\infty(b_n-1)\,\prod_{k=K}^{n-1}b_k$ ^ | + | $\prod_{k=K}^\infty b_k=\lim_{n\to\infty}a_n=\prod_{k=K}^{M-1}b_k+\sum_{n=M}^\infty(b_n-1)\,\prod_{k=K}^{n-1}b_k$ |
| + | |||
| + | ^ $\prod_{k=0}^\infty b_k = \prod_{k=0}^{M-1}b_k + \sum_{n=M}^\infty(b_n-1)\,\prod_{k=0}^{n-1}b_k$ ^ | ||
| <code> | <code> | ||
| b[n_] = 1 + 1/n!; | b[n_] = 1 + 1/n!; | ||
| + | |||
| + | Ks = 0; | ||
| + | Ke = 100; | ||
| + | Mid = 4; | ||
| + | |||
| + | Product[b[k], {k, Ks, Ke}] | ||
| + | |||
| Ks = 0; | Ks = 0; | ||
| Ke = 100; | Ke = 100; | ||
| Mid = 4; | Mid = 4; | ||
| - | Product[b[k], {k, Ks, Ke}] // N | ||
| Product[b[k], {k, Ks, Mid - 1}] + | Product[b[k], {k, Ks, Mid - 1}] + | ||
| - | Sum[(b[k] - 1) Product[b[k2], {k2, Ks, k - 1}], {k, Mid, Ke}] // N | + | Sum[(b[k] - 1) Product[b[k2], {k2, Ks, k - 1}], {k, Mid, Ke}] |
| </code> | </code> | ||
| Line 52: | Line 58: | ||
| >I've seen this pop up in the derivation of the basic path integral. Where else does it pop up? Why, if $x_n$ doesn't depend on $N$, does this turn to an $\exp(\int something)$? I think they join some limits to a single one. I think that relates to what Ron said about quantization procedures. | >I've seen this pop up in the derivation of the basic path integral. Where else does it pop up? Why, if $x_n$ doesn't depend on $N$, does this turn to an $\exp(\int something)$? I think they join some limits to a single one. I think that relates to what Ron said about quantization procedures. | ||
| + | |||
| + | <code> | ||
| + | h = (b - a)/m; | ||
| + | f[X_] = c X^2; | ||
| + | |||
| + | p = Product[ | ||
| + | 1 + f[i] h | ||
| + | , {i, a, b, h}] // simple | ||
| + | |||
| + | a = 0; | ||
| + | b = 7; | ||
| + | c = 3; | ||
| + | Limit[p, m -> \[Infinity]] | ||
| + | |||
| + | Exp[Integrate[c X^2, {X, 0, 7}]] | ||
| + | </code> | ||
| == Jacobi triple product == | == Jacobi triple product == | ||
| Line 61: | Line 83: | ||
| $\prod_{n=0}^{\infty}\left(1 + x^{2^n}\right) = \frac{1}{1-x}$ | $\prod_{n=0}^{\infty}\left(1 + x^{2^n}\right) = \frac{1}{1-x}$ | ||
| + | |||
| + | === Reference === | ||
| + | nLab: [[http://ncatlab.org/nlab/show/infinite+product|Infinite product]] | ||
| + | |||
| + | Wikipedia: [[http://en.wikipedia.org/wiki/Product_integral|Product integral]] | ||
| ----- | ----- | ||
| === Subset of === | === Subset of === | ||
| [[Infinite series]] | [[Infinite series]] | ||
