Jacobian theta functions with real arguments

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NAG[s21ccc] NAG[nag_jacobian_theta] - Jacobian theta functions with real arguments

Calling Sequence

s21ccc(k, x, q, 'fail'=fail)

nag_jacobian_theta(. . .)

Parameters

k - integer;

On entry: the function to be evaluated. Note that is equivalent to .

Constraint: . .

x - float;

On entry: the argument of the function.

Constraints:

x must not be an integer when and ;

must not be an integer when and ..

q - float;

On entry: the argument of the function.

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_jacobian_theta (s21ccc) returns the value of one of the Jacobian theta functions , , , or for a real argument and non-negative .

Description

nag_jacobian_theta (s21ccc) evaluates an approximation to the Jacobian theta functions , , , and given by

$[[theta[0](x,,,q),=1+2(∑)(-1)^nq^(n^2)cos(2,n,Pi,x) ],[theta[1](x,,,q),=2(∑)(-1)^nq^((n,+,1/2)^2)sin{(2,n,+,1),Pi,x} ],[theta[2](x,,,q),=2(∑)q^((n,+,1/2)^2)cos{(2,n,+,1),Pi,x} ],[theta[3](x,,,q),=1+2(∑)q^(n^2)cos(2,n,Pi,x) ],[theta[4](x,,,q),=theta[0](x,,,q) ]]$

where and (the nome) are real with . Note that is undefined if is an integer, as is if is an integer; otherwise, , for .

These functions are important in practice because every one of the Jacobian elliptic functions (see s21cbc (nag_jacobian_elliptic)) 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., Abramowitz and Stegun (1972), 16.27) 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

$theta[1](t,,,q)=2sqrt(lambda/Pi)e^(-lambdat^2)(∑)(-1)^ne^(-lambda(n,+,1/2)^2)sinh{(2,n,+,1),lambda,t}$

and is set equal to the first two terms (i.e., ) of

$theta[3](t,,,q)=sqrt(lambda/Pi)e^(-lambdat^2){1,+,2,(∑),e^(-lambdan^2),cosh(2,n,lambda,t)}$

where . Otherwise, the trigonometric series for and are used. For all values of , and are computed from the relations and .

	Error Indicators and Warnings
	"NE_INFINITE" The evaluation has been abandoned because the function value is infinite. "NE_INT" On entry, . Constraint: . "NE_INTERNAL_ERROR" An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please consult NAG for assistance. "NE_REAL" On entry, . Constraint: .

	Accuracy
	In principle nag_jacobian_theta (s21ccc) is capable of achieving full relative precision in the computed values. However, the accuracy obtainable in practice depends on the accuracy of the C standard library elementary functions such as sin and cos.