Timeline for Why does $d^n \exp(-x-x^{-1})/(dx)^n$ only have $n$ positive real zeroes?
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| Aug 23, 2015 at 8:42 | comment | added | Peter Mueller | Indeed, in order to approximately compute $\int_0^1\text{exp}(-x-1/x)dx$, Raabe (in 1836) computes approximately the four roots of $\phi_6(x)$ with $x>1$, and uses a theorem of Fourier (a refinement of Descartes for intervals) to show that there are no more roots. | |
| Aug 23, 2015 at 6:31 | comment | added | Benjamin Dickman | RE: (3) The $\phi_n$ appear - I don't think very helpfully, but do not read German - in an old paper of Raabe here. See II on page 95. | |
| Aug 21, 2015 at 19:35 | comment | added | Fedor Petrov | For $n=317$. That's awesome. Of course, it suffices for each $n$ to find and exponent $a_n$ such that the function $t^{a_n}\phi_{n+1}(t)/\phi_n(t)$ increases... Though this already looks less perspective than initial question. | |
| Aug 21, 2015 at 18:46 | history | edited | Peter Mueller | CC BY-SA 3.0 |
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| Aug 21, 2015 at 15:28 | history | edited | Peter Mueller | CC BY-SA 3.0 |
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| Aug 21, 2015 at 14:57 | history | answered | Peter Mueller | CC BY-SA 3.0 |