Jacobi's imaginary transformations relate elliptic functions to other elliptic functions of the same type but having different arguments. In the case of the Jacobi
elliptic functions, , and , the transformations are
and
is interpreted as satisfying (Whittaker and Watson 1990, p. 475).
Equation (6) can be written as the functional equation
(9)
where
and
is the half-period ratio (Davenport 1980, p. 62).
This form is useful for computing for small , since then the series for converges much faster than that for . In his paper of 1859, Riemann used this functional
equation for the theta function in one of his proofs of the functional equation for
the Riemann zeta function (Davenport 1980).
These transformations were first obtained by Jacobi (1828), but Poisson (1827) had previously obtained a formula equivalent to one of the four, and from which the other three follow from elementary algebra (Whittaker and Watson 1990, p. 475).
, 
, and 
, the transformations are
is the elliptic modulus, and 
is the complementary
modulus (Abramowitz and Stegun 1972; Whittaker and Watson 1990, p. 505).
is interpreted as satisfying 
(Whittaker and Watson 1990, p. 475).
and 
is the half-period ratio (Davenport 1980, p. 62).
This form is useful for computing 
for small 
, since then the series for 
converges much faster than that for 
. In his paper of 1859, Riemann used this functional
equation for the theta function in one of his proofs of the functional equation for
the Riemann zeta function (Davenport 1980).
