Gauß's constant

This article page is a stub, please help by expanding it.

Gauß’s constant is the reciprocal of the arithmetic-geometric mean of 1 and

2√  2

1 M  (1, 2√  2  ) ,

where

M  (a, b)

is the limit of the arithmetic-geometric mean iteration starting with

a

and

b

a 0 = a, b0 = b, an = an  − 1 + bn  − 1 2 , bn = 2√  an  − 1  bn  − 1 , n ≥ 1.

Carl Friedrich Gauß discovered (May 30, 1799) the following definite integral for this number

1 M  (1, 2√  2  )  =  2 π ∫ 1 0   1 2√  1 − x 4 d x.

Decimal expansion of Gauß’s constant

The decimal expansion of Gauß’s constant is

1 M  (1, 2√  2  )  =  0.8346268416740731862814297327990468...

giving the sequence of decimal digits (A014549)

{8, 3, 4, 6, 2, 6, 8, 4, 1, 6, 7, 4, 0, 7, 3, 1, 8, 6, 2, 8, 1, 4, 2, 9, 7, 3, 2, 7, 9, 9, 0, 4, 6, 8, 0, 8, 9, 9, 3, 9, 9, 3, 0, 1, 3, 4, 9, 0, 3, 4, 7, 0, 0, 2, 4, 4, 9, 8, ...}

Continued fraction for Gauß’s constant

The simple continued fraction for Gauß’s constant is

     

1 M  (1, 2√  2  )  =  1 1 +  1 5 +  1 21 +  1 3 +  1 4 +  1 14 +  1 ⋱

giving the sequence of integer part and partial quotients (A053002)

{0, 1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, 36, 1, 2, 5, 2, 1, 1, 2, 2, 6, 9, 1, 1, 1, 3, 1, 2, 6, 1, 5, 1, 1, 2, 1, 13, 2, 2, 5, 1, 2, 2, 1, 5, 1, 3, 1, 3, 1, 2, 2, ...}

Reciprocal of Gauß’s constant

Decimal expansion of reciprocal of Gauß’s constant

The decimal expansion of reciprocal of Gauß’s constant is

M  (1, 2√  2  )  =  1.19814023473559220743992249228...

giving the sequence of decimal digits (A053004)

{1, 1, 9, 8, 1, 4, 0, 2, 3, 4, 7, 3, 5, 5, 9, 2, 2, 0, 7, 4, 3, 9, 9, 2, 2, 4, 9, 2, 2, 8, 0, 3, 2, 3, 8, 7, 8, 2, 2, 7, 2, 1, 2, 6, 6, 3, 2, 1, 5, 6, 5, 1, 5, 5, 8, 2, ...}

Continued fraction for reciprocal of Gauß’s constant

The simple continued fraction for reciprocal of Gauß’s constant is

     

M  (1, 2√  2  )  =  1 +  1 5 +  1 21 +  1 3 +  1 4 +  1 14 +  1 ⋱  .

giving the sequence of integer part and partial quotients (A053003)

{1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, 36, 1, 2, 5, 2, 1, 1, 2, 2, 6, 9, 1, 1, 1, 3, 1, 2, 6, 1, 5, 1, 1, 2, 1, 13, 2, 2, 5, 1, 2, 2, 1, 5, 1, 3, 1, 3, 1, 2, 2, ...}