What do sin$(x)$ and arcsinh$(x)$ have in Common?
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910682301071360 |
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| author | Finch, Steven |
| author_facet | Finch, Steven |
| contents | N. G. de Bruijn (1958) studied the asymptotic expansion of iterates of sin$(x)$ with $0 < x \leq π/2$. Bencherif & Robin (1994) generalized this result to increasing analytic functions $f(x)$ with an attractive fixed point at 0 and $x > 0$ suitably small. Mavecha & Laohakosol (2013) formulated an algorithm for explicitly deriving required parameters. We review their method, testing it initally on the logistic function $\ell(x)$, a certain radical function $r(x)$, and later on several transcendental functions. Along the way, we show how $\ell(x)$ and $r(x)$ are kindred functions; the same is also true for sin$(x)$ and arcsinh$(x)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_01591 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | What do sin$(x)$ and arcsinh$(x)$ have in Common? Finch, Steven Classical Analysis and ODEs Discrete Mathematics Combinatorics 39A20 (Primary) 11B37, 26A18, 37E05, 41A60, 65D20 (Secondary) N. G. de Bruijn (1958) studied the asymptotic expansion of iterates of sin$(x)$ with $0 < x \leq π/2$. Bencherif & Robin (1994) generalized this result to increasing analytic functions $f(x)$ with an attractive fixed point at 0 and $x > 0$ suitably small. Mavecha & Laohakosol (2013) formulated an algorithm for explicitly deriving required parameters. We review their method, testing it initally on the logistic function $\ell(x)$, a certain radical function $r(x)$, and later on several transcendental functions. Along the way, we show how $\ell(x)$ and $r(x)$ are kindred functions; the same is also true for sin$(x)$ and arcsinh$(x)$. |
| title | What do sin$(x)$ and arcsinh$(x)$ have in Common? |
| topic | Classical Analysis and ODEs Discrete Mathematics Combinatorics 39A20 (Primary) 11B37, 26A18, 37E05, 41A60, 65D20 (Secondary) |
| url | https://arxiv.org/abs/2411.01591 |