Half-Period

An elliptic function can be characterized by its real and imaginary half-periods  and  (Whittaker and Watson 1990, p. 428), sometimes also denoted  (Abramowitz and Stegun 1972, p. 630). The Wolfram Languagecommand WeierstrassHalfPeriods[g2, g3] gives the half-periods  and  corresponding to the invariants and  for a Weierstrass elliptic function.

The notation

(1)

is sometimes also defined (Whittaker and Watson 1990, p. 443), although Abramowitz and Stegun (1972, p. 630) instead use the definition

(2)

In the case of a Weierstrass elliptic function, consider the modular discriminant

(3)

If , then  is real, and  is pure imaginary. However, if , then  is real and  is pure imaginary.

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 630, 1972.

Brezhnev, Y. V. "Uniformisation: On the Burnside Curve ." 9 Dec 2001. http://arxiv.org/abs/math.CA/0111150.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.