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Half-Period Ratio


tau is the ratio tau=omega_2/omega_1 of the two half-periods omega_1 and omega_2 of an elliptic function (Whittaker and Watson 1990, pp. 463 and 473) defined such that the imaginary part I[tau]>0. The notation t is sometimes used instead of tau.

The half-period ratio is most commonly encountered in the definition of the nome q as

 q=e^(piitau)
(1)

(Borwein and Borwein 1987, pp. 41, 109, and 114; Whittaker and Watson 1990, p. 463) where K(k) is the complete elliptic integral of the first kind, m=k^2 is the parameter, k is the elliptic modulus, K^'(k)=K(k^'), and k^' is the complementary elliptic modulus.

The notation

 x=-itau
(2)

is sometimes encountered in number theoretical literature (Davenport 1980, p. 62).

Unfortunately, in the theory of modular forms, it is common to instead define q=e^(2piitau). Care is therefore needed when consulting the literature. To avoid this ambiguity, it is therefore preferable to write

 q^_=q^2=e^(2piitau)
(3)

(Borwein and Borwein 1987, p. 118).


See also

Elliptic Invariants, Elliptic Modulus, Half-Period, Jacobi Theta Functions, Modular Angle, Inverse Nome, Nome, Parameter, Weierstrass Elliptic Function

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References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.Davenport, H. Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, 1980.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

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Half-Period Ratio

Cite this as:

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

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