Lemniscate Constant

Let

$$L={1\over\sqrt{2\pi}}[\Gamma({\textstyle{1\over 4}})]^2=5.2441151086\ldots$$

be the Arc Length of a Lemniscate with $a=1$. Then the lemniscate constant is the quantity $L/2$ (Abramowitz and Stegun 1972), or $L/4=1.311028777\ldots$ (Todd 1975, Le Lionnais 1983). Todd (1975) cites T. Schneider (1937) as proving $L$ to be a Transcendental Number.

See also Lemniscate

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/gauss/gauss.html

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 37, 1983.

Todd, J. ``The Lemniscate Constant.'' Comm. ACM 18, 14-19 and 462, 1975.