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 . 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.