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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_jactheta_real (s21cc)

## Purpose

nag_specfun_jactheta_real (s21cc) returns the value of one of the Jacobian theta functions ${\theta }_{0}\left(x,q\right)$, ${\theta }_{1}\left(x,q\right)$, ${\theta }_{2}\left(x,q\right)$, ${\theta }_{3}\left(x,q\right)$ or ${\theta }_{4}\left(x,q\right)$ for a real argument $x$ and non-negative $q<1$, via the function name.

## Syntax

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

## Description

nag_specfun_jactheta_real (s21cc) evaluates an approximation to the Jacobian theta functions ${\theta }_{0}\left(x,q\right)$, ${\theta }_{1}\left(x,q\right)$, ${\theta }_{2}\left(x,q\right)$, ${\theta }_{3}\left(x,q\right)$ and ${\theta }_{4}\left(x,q\right)$ given by
 $θ0x,q = 1+2∑n=1∞-1nqn2cos2nπx, θ1x,q = 2∑n=0∞-1nq n+12 2sin2n+1πx, θ2x,q = 2∑n=0∞q n+12 2cos2n+1πx, θ3x,q = 1+2∑n=1∞qn2cos2nπx, θ4x,q = θ0x,q,$
where $x$ and $q$ (the nome) are real with $0\le q<1$.
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 $\pi x$. 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 $x$ or $-x$ by an integer when $0\le t\le \frac{1}{2}$, it follows from the periodicity and symmetry properties of the functions that ${\theta }_{1}\left(x,q\right)=±{\theta }_{1}\left(t,q\right)$ and ${\theta }_{3}\left(x,q\right)=±{\theta }_{3}\left(t,q\right)$. In a region for which the approximation is sufficiently accurate, ${\theta }_{1}$ is set equal to the first term ($n=0$) of the transformed series
 $θ1t,q=2λπe-λt2∑n=0∞-1ne-λ n+12 2sinh2n+1λt$
and ${\theta }_{3}$ is set equal to the first two terms (i.e., $n\le 1$) of
 $θ3t,q=λπe-λt2 1+2∑n=1∞e-λn2cosh2nλt ,$
where $\lambda ={\pi }^{2}/\left|{\mathrm{log}}_{\mathrm{e}}q\right|$. Otherwise, the trigonometric series for ${\theta }_{1}\left(t,q\right)$ and ${\theta }_{3}\left(t,q\right)$ are used. For all values of $x$, ${\theta }_{0}$ and ${\theta }_{2}$ are computed from the relations ${\theta }_{0}\left(x,q\right)={\theta }_{3}\left(\frac{1}{2}-\left|x\right|,q\right)$ and ${\theta }_{2}\left(x,q\right)={\theta }_{1}\left(\frac{1}{2}-\left|x\right|,q\right)$.

## 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:     $\mathrm{k}$int64int32nag_int scalar
Denotes the function ${\theta }_{k}\left(x,q\right)$ to be evaluated. Note that ${\mathbf{k}}=4$ is equivalent to ${\mathbf{k}}=0$.
Constraint: $0\le {\mathbf{k}}\le 4$.
2:     $\mathrm{x}$ – double scalar
The argument $x$ of the function.
3:     $\mathrm{q}$ – double scalar
The argument $q$ of the function.
Constraint: $0.0\le {\mathbf{q}}<1.0$.

None.

### Output Parameters

1:     $\mathrm{result}$ – double scalar
The result of the function.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ 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.

${\mathbf{ifail}}=1$
 On entry, ${\mathbf{k}}<0$, or ${\mathbf{k}}>4$, or ${\mathbf{q}}<0.0$, or ${\mathbf{q}}\ge 1.0$,
W  ${\mathbf{ifail}}=2$
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)).
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
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.

None.

## Example

This example evaluates ${\theta }_{2}\left(x,q\right)$ at $x=0.7$ when $q=0.4$, and prints the results.
```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
```