Half-Period Ratio
المؤلف:
Borwein, J. M. and Borwein, P.
المصدر:
Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.
الجزء والصفحة:
...
22-4-2019
2232
Half-Period Ratio
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 ![I[tau]>0](http://mathworld.wolfram.com/images/equations/Half-PeriodRatio/Inline5.gif)
. 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).
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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