A309948 Decimal expansion of the real part of the square root of 1 + i.

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, 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, 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

Offset

1,3

Comments

i is the imaginary unit such that i^2 = -1.

Also imaginary part of sqrt(-1 + i).

Links

Table of n, a(n) for n=1..96.

Jean-Paul Allouche, Samin Riasat, and Jeffrey Shallit, More infinite products: Thue-Morse and the Gamma function, The Ramanujan Journal, Vol. 49 (2019), pp. 115-128; arXiv preprint, arXiv:1709.03398 [math.NT], 2017.

Formula

Re(sqrt(1 + i)) = sqrt(1/2 + 1/sqrt(2)) = 2^(1/4) * cos(Pi/8).

Equals Im(-sqrt(-1 - i)). - Peter Luschny, Sep 20 2019

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

Example

Re(sqrt(1 + i)) = 1.09868411346780996603980119524...

Maple

Digits := 120: Im(-sqrt(-1 - I))*10^95:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Sep 20 2019

Mathematica

RealDigits[Sqrt[1/2 + 1/Sqrt[2]], 10, 100][[1]]

Prog

(PARI) real(sqrt(1+I)) \\ Michel Marcus, Sep 16 2019

Crossrefs

Cf. A010060, A309949 (imaginary part).

Sequence in context: A155920 A082124 A132721 * A086053 A358661 A129269

Adjacent sequences: A309945 A309946 A309947 * A309949 A309950 A309951

Keyword

nonn,cons,easy

Author

Alonso del Arte, Aug 24 2019

Status

approved