A175573 Decimal expansion of Pi^(1/4)/Gamma(3/4).
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, 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, 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
Offset: 1
Examples
1.0864348112133080145753161...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..5000
- Bruce C. Berndt, Chapter 11 of Ramanujan's second notebook, Bull. Lond. Math. Soc., Vol. 15, No. 4 (1983), 273-320.
- Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 27, equation 27:13:18 at page 258.
- Wikipedia, Theta function.
Crossrefs
Programs
-
Magma
C := ComplexField(); [(Pi(C))^(1/4)/Gamma(3/4)]; // G. C. Greubel, Nov 05 2017
-
Maple
Pi^(1/4)/GAMMA(3/4) ; evalf(%) ;
-
Mathematica
RealDigits[ Pi^(1/4)/Gamma[3/4], 10, 105][[1]] (* Jean-François Alcover, Jul 04 2013 *)
-
PARI
Pi^(1/4)/gamma(3/4) \\ G. C. Greubel, Nov 05 2017
-
PARI
2*suminf(k=0,exp(-Pi)^(k^2))-1 \\ Hugo Pfoertner, Sep 17 2018
Formula
Equals Sum_{n=-oo..oo} exp(-Pi*n^2), or also EllipticTheta(3, 0, exp(-Pi)). - Jean-François Alcover, Jul 04 2013
Equals sqrt(A175574). - Amiram Eldar, Jul 04 2023
Equals Gamma(1/4)/(sqrt(2)*Pi^(3/4)). - Vaclav Kotesovec, Jul 04 2023
Equals Product_{k>=1} tanh((1/2 + i/2)*Pi*k), i=sqrt(-1). - _Antonio Graciá Llorente, Mar 20 2024
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
Equals (1/A096427)^2 (see Spanier and Oldham at p. 258). - Stefano Spezia, Dec 31 2024
Comments