is the ratio 
of the two half-periods 
and 
of an elliptic function
(Whittaker and Watson 1990, pp. 463 and 473) defined such that the imaginary
part 
.
The notation 
is sometimes used instead of 
.
The half-period ratio is most commonly encountered in the definition of the nome 
as
 |
(1) |
(Borwein and Borwein 1987, pp. 41, 109, and 114; Whittaker and Watson 1990, p. 463) where 
is the complete elliptic integral
of the first kind, 
is the parameter, 
is the elliptic modulus,
,
and 
is the complementary elliptic modulus.
The notation
 |
(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 
. Care is therefore needed
when consulting the literature. To avoid this ambiguity, it is therefore preferable
to write
 |
(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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Cite this as:
Weisstein, Eric W. "Half-Period Ratio." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Half-PeriodRatio.html