3 in 1: a simple way to prove that \(e^r\), \(\mathrm{ln}(r)\) and \(\pi^2\) are irrational (Q670728)
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scientific article; zbMATH DE number 7039092
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | 3 in 1: a simple way to prove that \(e^r\), \(\mathrm{ln}(r)\) and \(\pi^2\) are irrational |
scientific article; zbMATH DE number 7039092 |
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3 in 1: a simple way to prove that \(e^r\), \(\mathrm{ln}(r)\) and \(\pi^2\) are irrational (English)
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20 March 2019
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The main result of tis paper is that if \(J_n(z)=z^{n+1}\int _0^1 \frac 1{n!} (t^n(1-t)^n)^{(n)} e^{zt}dt\) does not eventualy vanish then not both \(z\) and \(e^z\) can be Gaussian rationals. The proof makes use the Padé approximation.
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irrationality
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Pade's approximation
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number e
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0.88177013
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0.87458915
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0.8671888
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0.8656966
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0.8541737
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