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Half-Period

An elliptic function can be characterized by its real and imaginary half-periods omega_1 and omega_2 (Whittaker and Watson 1990, p. 428), sometimes also denoted (omega,omega^') (Abramowitz and Stegun 1972, p. 630). The Wolfram Language command WeierstrassHalfPeriods[{g2, g3}] gives the half-periods omega and omega^' corresponding to the invariants g_2 and g_3 for a Weierstrass elliptic function.

The notation

omega_3=-omega_1-omega_2
(1)

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

omega_3=omega_2-omega_1.
(2)

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

Delta=g_2^3-27g_3^2.
(3)

If Delta<0, then omega_2 is real, and omega_2^'=omega^'-omega is pure imaginary. However, if Delta>0, then omega is real and omega_2^'=omega^'-omega is pure imaginary.

See also

Elliptic Invariants, Half-Period Ratio, omega2 Constant, Weierstrass Elliptic Function

Related Wolfram sites

http://functions.wolfram.com/EllipticFunctions/WeierstrassHalfPeriods/, http://functions.wolfram.com/EllipticFunctions/WeierstrassPHalfPeriodValues/

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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 y^2=x^5-x." 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.

Referenced on Wolfram|Alpha

Half-Period

Cite this as:

Weisstein, Eric W. "Half-Period." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Half-Period.html

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