Gauss's constant

The lemniscate is a constant introduced by Carl Friedrich Gauss in 1798 mathematical constant. It is defined as the value of the elliptic integral

And occurs in the calculation of the arc length of the entire lemniscate.

Designation

Gauss chose for the lemniscate constant aware of the Greek minuscule (pronounced Pi - script or Varpi ), an alternative spelling of, to the analogy to the circle with half its circumference

To remember. The origin of this designation at Gauss cleared probably first Ludwig Schlesinger on: Gauss Initially used to designate the lemniscate period, the characters, and from July 1798 he used consistently for that size.

In English, there is something for the minuscule, the ( misleading ) name pomega.

In the English -speaking world is

Called Gaußkonstante.

Properties

Applies with the beta function and the gamma function

Gauss found the relationship

With the arithmetic- geometric mean and fast converging series was also a

With summands of the order. The evaluation

The elliptic integral yields a similar series, but much more slowly converges, since the members of the order are. Very quickly, the series converges in

With summands of the order.

Niels Nielsen introduced in 1906 with the help of Kummer's series of the gamma function is a connection with the Euler constant here:

Theodor Schneider proved the transcendence of 1937. Gregory Chudnovsky showed in 1975 that and thus is also algebraically independent of.