nag_jacobian_theta (s21ccc) returns the value of one of the Jacobian theta functions , , , or for a real argument and non-negative .
nag_jacobian_theta (s21ccc) evaluates an approximation to the Jacobian theta functions
,
,
,
and
given by
where
and
(the
nome) are real with
.
These functions are important in practice because every one of the Jacobian elliptic functions (see
nag_jacobian_elliptic (s21cbc)) can be expressed as the ratio of two Jacobian theta functions (see
Whittaker and Watson (1990)). There is also a bewildering variety of notations used in the literature to define them. Some authors (e.g., Section 16.27 of
Abramowitz and Stegun (1972)) define the argument in the trigonometric terms to be
instead of
. This can often lead to confusion, so great care must therefore be exercised when consulting the literature. Further details (including various relations and identities) can be found in the references.
nag_jacobian_theta (s21ccc) is based on a truncated series approach. If
differs from
or
by an integer when
, it follows from the periodicity and symmetry properties of the functions that
and
. In a region for which the approximation is sufficiently accurate,
is set equal to the first term (
) of the transformed series
and
is set equal to the first two terms (i.e.,
) of
where
. Otherwise, the trigonometric series for
and
are used. For all values of
,
and
are computed from the relations
and
.
In principle the function is capable of achieving full relative precision in the computed values. However, the accuracy obtainable in practice depends on the accuracy of the standard elementary functions such as sin and cos.
Not applicable.
None.