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<a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

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Equals exp(5*Pi/24) * sqrt(2 + sqrt(5) - sqrt((15 + 7*sqrt(5))/2))/2^(1/8). - Vaclav Kotesovec, May 13 2023

RealDigits[Exp[5*Pi/24]*Sqrt[2 + Sqrt[5] - Sqrt[(15 + 7*Sqrt[5])/2]]/2^(1/8), 10, 120][[1]] (* Vaclav Kotesovec, May 13 2023 *)

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(PARI) polrootsreal(2^(3/4)*'x^6 + 2^(17/8)*exp(5*Pi/24)*'x^5 + 2^(5/8)*exp(25*Pi/24)*'x - exp(5*Pi/4))[2] \\ Charles R Greathouse IV, Mar 04 2018

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allocated for Vaclav KotesovecDecimal expansion of Product_{k>=1} (1 + exp(-5*Pi*k)).

1, 0, 0, 0, 0, 0, 0, 1, 5, 0, 7, 0, 1, 7, 5, 0, 2, 5, 0, 0, 2, 3, 9, 8, 9, 4, 9, 3, 8, 6, 9, 8, 7, 1, 4, 6, 7, 9, 7, 3, 7, 6, 1, 0, 0, 6, 4, 3, 0, 5, 0, 7, 4, 0, 5, 6, 9, 0, 1, 9, 9, 9, 8, 8, 5, 2, 0, 8, 8, 7, 1, 3, 4, 4, 2, 6, 9, 4, 9, 7, 1, 7, 6, 1, 8, 7, 2, 8, 7, 4, 6, 7, 3, 2, 5, 8, 5, 1, 0, 0, 2, 8, 5, 0, 4

1,9

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DedekindEtaFunction.html">Dedekind Eta Function</a>

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>

Wikipedia, <a href="https://en.wikipedia.org/wiki/Dedekind_eta_function">Dedekind eta function</a>

Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler_function">Euler function</a>

Root r of the equation 2^(3/4)*r^6 + 2^(17/8)*exp(5*Pi/24)*r^5 + 2^(5/8)*exp(25*Pi/24)*r - exp(5*Pi/4) = 0.

1.000000150701750250023989493869871467973761006430507405690199988520887...

RealDigits[r/.FindRoot[2^(3/4)*r^6 + 2^(17/8)*E^(5*Pi/24)*r^5 + 2^(5/8)*E^(25*Pi/24)*r - E^(5*Pi/4) == 0, {r, 1}, WorkingPrecision -> 120], 10, 120][[1]]

RealDigits[QPochhammer[-1, E^(-5*Pi)]/2, 10, 120][[1]]

Cf. A292819, A292820, A292821, A292887, A292822, A292905.

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Vaclav Kotesovec, Sep 26 2017

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