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A273082 Decimal expansion of theta_3(0, exp(-4*Pi)), where theta_3 is the 3rd Jacobi theta function.

%I A273082 #17 May 17 2023 04:33:15
%S A273082 1,0,0,0,0,0,6,9,7,4,6,8,4,7,1,2,4,1,7,9,9,1,2,7,9,3,5,7,4,5,5,7,2,2,
%T A273082 7,7,3,3,8,6,0,8,4,8,1,1,8,1,9,3,4,3,9,5,9,6,7,0,2,4,3,4,2,3,6,2,3,8,
%U A273082 8,2,3,7,0,8,1,9,5,5,9,4,5,4,9,6,1,9,2,5,3,0,0,9,2,4,6,2,9,9,5,1,4,6,7,9,7
%N A273082 Decimal expansion of theta_3(0, exp(-4*Pi)), where theta_3 is the 3rd Jacobi theta function.
%H A273082 G. C. Greubel, <a href="/A273082/b273082.txt">Table of n, a(n) for n = 1..10000</a>
%H A273082 Wikipedia, <a href="http://en.wikipedia.org/wiki/Theta_function#Explicit_values">Theta function</a>
%F A273082 Equals (2 + 2^(3/4))/4 * Pi^(1/4) / Gamma(3/4).
%F A273082 Equals (1 + 2^(1/4)) * Gamma(1/4) / (2^(7/4)*Pi^(3/4)).
%F A273082 Equals sqrt((A247217^2 + A259149^4/A259150^2)/2). - _Vaclav Kotesovec_, May 17 2023
%e A273082 1.00000697468471241799127935745572277338608481181934395967...
%p A273082 evalf((2+2^(3/4))/4*Pi^(1/4)/GAMMA(3/4), 120);
%p A273082 evalf((1 + 2^(1/4)) * GAMMA(1/4) / (2^(7/4)*Pi^(3/4)), 120);
%t A273082 RealDigits[EllipticTheta[3, 0, Exp[-4*Pi]], 10, 105][[1]]
%t A273082 RealDigits[(2 + 2^(3/4))/4 * Pi^(1/4) / Gamma[3/4], 10, 105][[1]]
%t A273082 RealDigits[(1 + 2^(1/4)) * Gamma[1/4] / (2^(7/4)*Pi^(3/4)), 10, 105][[1]]
%o A273082 (PARI) (2^.25+1)*gamma(1/4)/sqrtn(128*Pi^3,4) \\ _Charles R Greathouse IV_, Jun 06 2016
%o A273082 (Magma) C<i> := ComplexField(); ((2+2^(3/4))/4)*Pi(C)^(1/4)/Gamma(3/4) // _G. C. Greubel_, Jan 07 2018
%Y A273082 Cf. A175573, A247217, A273081, A273083, A273084.
%K A273082 nonn,cons
%O A273082 1,7
%A A273082 _Vaclav Kotesovec_, May 14 2016