These functions are important in practice because every one of the Jacobian elliptic functions (see nag_specfun_jacellip_complex (s21cb)) 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

π x

. 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_specfun_jactheta_real (s21cc) is based on a truncated series approach. If

t

differs from

x

- x

by an integer when

0 \leq t \leq \frac{1}{2}

, it follows from the periodicity and symmetry properties of the functions that

θ_{1} (x, q) = \pm θ_{1} (t, q)

and

θ_{3} (x, q) = \pm θ_{3} (t, q)

. In a region for which the approximation is sufficiently accurate,

θ_{1}

is set equal to the first term (

n = 0

) of the transformed series

θ_{1} (t, q) = 2 \sqrt{\frac{λ}{π}} e^{- λ t^{2}} \sum_{n = 0}^{\infty} {(- 1)}^{n} e^{- λ {(n + \frac{1}{2})}^{2}} \sinh \{(2 n + 1) λ t\}

and

θ_{3}

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

n \leq 1

) of

θ_{3} (t, q) = \sqrt{\frac{λ}{π}} e^{- λ t^{2}} \{1 + 2 \sum_{n = 1}^{\infty} e^{- λ n^{2}} \cosh (2 n λ t)\},

where

λ = π^{2} / |\log_{e} q|

. Otherwise, the trigonometric series for

θ_{1} (t, q)

and

θ_{3} (t, q)

are used. For all values of

x

θ_{0}

and

θ_{2}

are computed from the relations

θ_{0} (x, q) = θ_{3} (\frac{1}{2} - |x|, q)

and

θ_{2} (x, q) = θ_{1} (\frac{1}{2} - |x|, q)

References

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

Parameters

Compulsory Input Parameters

1: $k$ – int64int32nag_int scalar: Denotes the function $θ_{k} (x, q)$ to be evaluated. Note that $k = 4$ is equivalent to $k = 0$ .

Constraint: $0 \leq k \leq 4$ .
2: $x$ – double scalar: The argument $x$ of the function.
3: $q$ – double scalar: The argument $q$ of the function.

Constraint: $0.0 \leq q < 1.0$ .

Optional Input Parameters

None.

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$

On entry,	$k < 0$ ,
or	$k > 4$ ,
or	$q < 0.0$ ,
or	$q \geq 1.0$ ,

W $ifail = 2$: The evaluation has been abandoned because the function value is infinite. The result is returned as the largest machine representable number (see nag_machine_real_largest (x02al)).

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

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.

Further Comments

None.

Example

This example evaluates

θ_{2} (x, q)

x = 0.7

when

q = 0.4

, and prints the results.

Open in the MATLAB editor: s21cc_example

function s21cc_example


fprintf('s21cc example results\n\n');

k = int64(2);
x = 0.7;
q = 0.4;

[result, ifail] = s21cc(k, x, q);

fprintf('%3s%7s%7s%16s\n','k','x','q','theta_k(x,q)');
fprintf('%3d  %7.2f%7.2f%14.4e\n',k,x,q,result);

s21cc example results

  k      x      q    theta_k(x,q)
  2     0.70   0.40   -6.9289e-01

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox