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infinite_product_of_complex_numbers [2015/04/16 13:21] nikolaj |
infinite_product_of_complex_numbers [2016/04/02 14:53] (current) nikolaj |
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>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 28: | 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> | ||