# NAG CL Interfaces21ccc (jactheta_​real)

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## 1Purpose

s21ccc 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 $q<1$.

## 2Specification

 #include
 double s21ccc (Integer k, double x, double q, NagError *fail)
The function may be called by the names: s21ccc, nag_specfun_jactheta_real or nag_jacobian_theta.

## 3Description

s21ccc 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
 $θ0(x,q) = 1+2∑n=1∞(-1)nqn2cos(2nπx), θ1(x,q) = 2∑n=0∞(-1)nq (n+12) 2sin{(2n+1)πx}, θ2(x,q) = 2∑n=0∞q (n+12) 2cos{(2n+1)πx}, θ3(x,q) = 1+2∑n=1∞qn2cos(2nπx), θ4(x,q) = θ0(x,q),$
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 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 $\pi x$. 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.
s21ccc 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
 $θ1(t,q)=2λπe-λt2∑n=0∞(-1)ne-λ (n+12) 2sinh{(2n+1)λt}$
and ${\theta }_{3}$ is set equal to the first two terms (i.e., $n\le 1$) of
 $θ3(t,q)=λπe-λt2 {1+2∑n=1∞e-λn2cosh(2nλt)} ,$
where $\lambda ={\pi }^{2}/|{\mathrm{log}}_{\mathrm{e}}q|$. 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 ${\theta }_{0}\left(x,q\right)={\theta }_{3}\left(\frac{1}{2}-|x|,q\right)$ and ${\theta }_{2}\left(x,q\right)={\theta }_{1}\left(\frac{1}{2}-|x|,q\right)$.

## 4References

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

## 5Arguments

1: $\mathbf{k}$Integer Input
On entry: 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: $\mathbf{x}$double Input
On entry: the argument $x$ of the function.
3: $\mathbf{q}$double Input
On entry: the argument $q$ of the function.
Constraint: $0.0\le {\mathbf{q}}<1.0$.
4: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_INT
On entry, ${\mathbf{k}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{k}}\le 4$.
On entry, ${\mathbf{k}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{k}}\ge 0$.
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 contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, ${\mathbf{q}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{q}}<1.0$.
On entry, ${\mathbf{q}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{q}}\ge 0.0$.

## 7Accuracy

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.

## 8Parallelism and Performance

s21ccc is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (s21ccce.c)

### 10.2Program Data

Program Data (s21ccce.d)

### 10.3Program Results

Program Results (s21ccce.r)