|
%I #22 Feb 04 2024 03:25:27
%S 1,0,9,8,6,8,4,1,1,3,4,6,7,8,0,9,9,6,6,0,3,9,8,0,1,1,9,5,2,4,0,6,7,8,
%T 3,7,8,5,4,4,3,9,3,1,2,0,9,2,7,1,5,7,7,4,3,7,4,4,4,1,1,5,7,8,8,4,2,8,
%U 7,5,0,5,3,5,5,5,2,8,4,8,1,1,1,3,6,5,3,6,0,6,6,3,5,6,4,1
%N Decimal expansion of the real part of the square root of 1 + i.
%C i is the imaginary unit such that i^2 = -1.
%C Also imaginary part of sqrt(-1 + i).
%H Jean-Paul Allouche, Samin Riasat, and Jeffrey Shallit, <a href="https://doi.org/10.1007/s11139-017-9981-7">More infinite products: Thue-Morse and the Gamma function</a>, The Ramanujan Journal, Vol. 49 (2019), pp. 115-128; <a href="https://arxiv.org/abs/1709.03398">arXiv preprint</a>, arXiv:1709.03398 [math.NT], 2017.
%F Re(sqrt(1 + i)) = sqrt(1/2 + 1/sqrt(2)) = 2^(1/4) * cos(Pi/8).
%F Equals Im(-sqrt(-1 - i)). - _Peter Luschny_, Sep 20 2019
%F Equals Product_{k>=0} ((8*k+3)*(8*k+5)/((8*k+1)*(8*k+7)))^A010060(k) (Allouche et al., 2019). - _Amiram Eldar_, Feb 04 2024
%e Re(sqrt(1 + i)) = 1.09868411346780996603980119524...
%p Digits := 120: Im(-sqrt(-1 - I))*10^95:
%p ListTools:-Reverse(convert(floor(%), base, 10)); # _Peter Luschny_, Sep 20 2019
%t RealDigits[Sqrt[1/2 + 1/Sqrt[2]], 10, 100][[1]]
%o (PARI) real(sqrt(1+I)) \\ _Michel Marcus_, Sep 16 2019
%Y Cf. A010060, A309949 (imaginary part).
%K nonn,cons,easy
%O 1,3
%A _Alonso del Arte_, Aug 24 2019
|
