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A175573 Decimal expansion of Pi^(1/4)/Gamma(3/4).

%I A175573 #43 Jan 01 2025 00:14:46
%S A175573 1,0,8,6,4,3,4,8,1,1,2,1,3,3,0,8,0,1,4,5,7,5,3,1,6,1,2,1,5,1,0,2,2,3,
%T A175573 4,5,7,0,7,0,2,0,5,7,0,7,2,4,5,2,1,8,8,8,5,9,2,0,7,9,0,3,1,5,9,8,1,8,
%U A175573 5,6,7,3,2,2,6,7,1,0,9,7,9,5,9,6,0,5,6,1,6,1,8,4,8,9,6,7,9,7,6,4,0,3,7,4,1
%N A175573 Decimal expansion of Pi^(1/4)/Gamma(3/4).
%C A175573 Entry 34 a of chapter 11 of Ramanujan's second notebook. Entry 34 b is A085565.
%H A175573 G. C. Greubel, <a href="/A175573/b175573.txt">Table of n, a(n) for n = 1..5000</a>
%H A175573 Bruce C. Berndt, <a href="http://dx.doi.org/10.1112/blms/15.4.273">Chapter 11 of Ramanujan's second notebook</a>, Bull. Lond. Math. Soc., Vol. 15, No. 4 (1983), 273-320.
%H A175573 Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 27, equation 27:13:18 at page 258.
%H A175573 Wikipedia, <a href="http://en.wikipedia.org/wiki/Theta_function#Explicit_values">Theta function</a>.
%F A175573 Equals A092040 / A068465.
%F A175573 Equals Sum_{n=-oo..oo} exp(-Pi*n^2), or also EllipticTheta(3, 0, exp(-Pi)). - _Jean-François Alcover_, Jul 04 2013
%F A175573 Equals sqrt(A175574). - _Amiram Eldar_, Jul 04 2023
%F A175573 Equals Gamma(1/4)/(sqrt(2)*Pi^(3/4)). - _Vaclav Kotesovec_, Jul 04 2023
%F A175573 Equals Product_{k>=1} tanh((1/2 + i/2)*Pi*k), i=sqrt(-1). - __Antonio Graciá Llorente_, Mar 20 2024
%F A175573 Equals Product_{k>=0} (1/2)*(((k+1/2)/(k+1))^(1/2)+((k+1)/(k+1/2))^(1/2)). - _Antonio Graciá Llorente_, Jul 23 2024
%F A175573 Equals (1/A096427)^2 (see Spanier and Oldham at p. 258). - _Stefano Spezia_, Dec 31 2024
%F A175573 Equals 2*A319332 = 1/A327995. - _Hugo Pfoertner_, Dec 31 2024
%e A175573 1.0864348112133080145753161...
%p A175573 Pi^(1/4)/GAMMA(3/4) ; evalf(%) ;
%t A175573 RealDigits[ Pi^(1/4)/Gamma[3/4], 10, 105][[1]] (* _Jean-François Alcover_, Jul 04 2013 *)
%o A175573 (PARI) Pi^(1/4)/gamma(3/4) \\ _G. C. Greubel_, Nov 05 2017
%o A175573 (PARI) 2*suminf(k=0,exp(-Pi)^(k^2))-1 \\ _Hugo Pfoertner_, Sep 17 2018
%o A175573 (Magma) C<i> := ComplexField(); [(Pi(C))^(1/4)/Gamma(3/4)]; // _G. C. Greubel_, Nov 05 2017
%Y A175573 Cf. A068465, A085565, A092040, A096427, A175574, A247217, A273081, A273082, A273083, A273084, A319332, A327995.
%K A175573 cons,easy,nonn
%O A175573 1,3
%A A175573 _R. J. Mathar_, Jul 15 2010