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# Gauß's constant

(Redirected from Gauss's constant)

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
dx.

## 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, ...}