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](/images/equations/Half-PeriodRatio/Inline5.svg)
.
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).
