NAG Library Function Document

nag_jacobian_theta (s21ccc)


    1  Purpose
    7  Accuracy


nag_jacobian_theta (s21ccc) returns the value of one of the Jacobian theta functions θ0x,q, θ1x,q, θ2x,q, θ3x,q or θ4x,q for a real argument x and non-negative q<1.


#include <nag.h>
#include <nags.h>
double  nag_jacobian_theta (Integer k, double x, double q, NagError *fail)


nag_jacobian_theta (s21ccc) evaluates an approximation to the Jacobian theta functions θ0x,q, θ1x,q, θ2x,q, θ3x,q and θ4x,q given by
θ0x,q = 1+2n=1-1nqn2cos2nπx, θ1x,q = 2n=0-1nq n+12 2sin2n+1πx, θ2x,q = 2n=0q n+12 2cos2n+1πx, θ3x,q = 1+2n=1qn2cos2nπx, θ4x,q = θ0x,q,  
where x and q (the nome) are real with 0q<1.
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 x instead of πx. 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 t differs from x or -x by an integer when 0t12 , it follows from the periodicity and symmetry properties of the functions that θ1x,q=±θ1t,q and θ3x,q=±θ3t,q. In a region for which the approximation is sufficiently accurate, θ1 is set equal to the first term (n=0) of the transformed series
θ1t,q=2λπe-λt2n=0-1ne-λ n+12 2sinh2n+1λt  
and θ3 is set equal to the first two terms (i.e., n1) of
θ3t,q=λπe-λt2 1+2n=1e-λn2cosh2nλt ,  
where λ= π2/ logeq . Otherwise, the trigonometric series for θ1t,q and θ3t,q are used. For all values of x, θ0 and θ2 are computed from the relations θ0x,q=θ312-x,q and θ2x,q=θ112-x,q.


Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Byrd P F and Friedman M D (1971) Handbook of Elliptic Integrals for Engineers and Scientists pp. 315–320 (2nd Edition) Springer–Verlag
Magnus W, Oberhettinger F and Soni R P (1966) Formulas and Theorems for the Special Functions of Mathematical Physics 371–377 Springer–Verlag
Tølke F (1966) Praktische Funktionenlehre (Bd. II) 1–38 Springer–Verlag
Whittaker E T and Watson G N (1990) A Course in Modern Analysis (4th Edition) Cambridge University Press


1:     k IntegerInput
On entry: denotes the function θkx,q to be evaluated. Note that k=4 is equivalent to k=0.
Constraint: 0k4.
2:     x doubleInput
On entry: the argument x of the function.
3:     q doubleInput
On entry: the argument q of the function.
Constraint: 0.0q<1.0.
4:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, k=value.
Constraint: k4.
On entry, k=value.
Constraint: k0.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
On entry, q=value.
Constraint: q<1.0.
On entry, q=value.
Constraint: q0.0.


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.

Parallelism and Performance

nag_jacobian_theta (s21ccc) is not threaded in any implementation.

Further Comments



This example evaluates θ2x,q at x=0.7 when q=0.4, and prints the results.

Program Text

Program Text (s21ccce.c)

Program Data

Program Data (s21ccce.d)

Program Results

Program Results (s21ccce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017