naginterfaces.library.specfun.jactheta_​real

naginterfaces.library.specfun.jactheta_real(k, x, q)

jactheta_real returns the value of one of the Jacobian theta functions ${\theta }_0(x,q)$, ${\theta }_1(x,q)$, ${\theta }_2(x,q)$, ${\theta }_3(x,q)$ or ${\theta }_4(x,q)$ for a real argument $x$ and non-negative $q<1$.

For full information please refer to the NAG Library document for s21cc

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s21ccf.html

Parameters

kint

Denotes the function ${\theta }_k(x,q)$ to be evaluated. Note that $k=4$ is equivalent to $k=0$.

xfloat

The argument $x$ of the function.

qfloat

The argument $q$ of the function.

Returns

jtfloat

The value of one of the Jacobian theta functions ${\theta }_0(x,q)$, ${\theta }_1(x,q)$, ${\theta }_2(x,q)$, ${\theta }_3(x,q)$ or ${\theta }_4(x,q)$ for a real argument $x$ and non-negative $q<1$.

Raises

NagValueError

(errno $1$)

On entry, $q=⟨value⟩$.

Constraint: $q<1.0$.

(errno $1$)

On entry, $q=⟨value⟩$.

Constraint: $q\ge 0.0$.

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k\le 4$.

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 0$.

Notes

jactheta_real evaluates an approximation to the Jacobian theta functions ${\theta }_0(x,q)$, ${\theta }_1(x,q)$, ${\theta }_2(x,q)$, ${\theta }_3(x,q)$ and ${\theta }_4(x,q)$ given by

$$\begin{array}{c}\begin{array}{ccc}{\theta }_0(x,q)& =& 1+2{\sum }_{n=1}^{\infty }{(-1)}^n{q}^{n^2}\cos \left(2n\pi x\right)\text{,}\\ & & \\ {\theta }_1(x,q)& =& 2{\sum }_{n=0}^{\infty }{(-1)}^n{q}^{{(n+\frac{1}{2})}^2}\sin \left\{(2n+1)\pi x\right\}\text{,}\\ & & \\ {\theta }_2(x,q)& =& 2{\sum }_{n=0}^{\infty }{q}^{{(n+\frac{1}{2})}^2}\cos \left\{(2n+1)\pi x\right\}\text{,}\\ & & \\ {\theta }_3(x,q)& =& 1+2{\sum }_{n=1}^{\infty }{q}^{n^2}\cos \left(2n\pi x\right)\text{,}\\ & & \\ {\theta }_4(x,q)& =& {\theta }_0(x,q)\text{,}\end{array}\end{array}$$

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 jacellip_complex()) 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.

jactheta_real 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 ${\theta }_1(x,q)=\pm {\theta }_1(t,q)$ and ${\theta }_3(x,q)=\pm {\theta }_3(t,q)$. 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

$${\theta }_1(t,q)=2\sqrt{\frac{\lambda }{\pi }}{e}^{-\lambda t^2}\sum _{n=0}^{\infty }{(-1)}^n{e}^{-\lambda {(n+\frac{1}{2})}^2}\sinh \left\{(2n+1)\lambda t\right\}$$

and ${\theta }_3$ is set equal to the first two terms (i.e., $n\le 1$) of

$${\theta }_3(t,q)=\sqrt{\frac{\lambda }{\pi }}{e}^{-\lambda t^2}\{1+2\sum _{n=1}^{\infty }{e}^{-\lambda n^2}\cosh \left(2n\lambda t\right)\}\text{,}$$

where $\lambda ={\pi }^2/\left|loge\left(q\right)\right|$. Otherwise, the trigonometric series for ${\theta }_1(t,q)$ and ${\theta }_3(t,q)$ are used. For all values of $x$, ${\theta }_0$ and ${\theta }_2$ are computed from the relations ${\theta }_0(x,q)={\theta }_3(\frac{1}{2}-\left|x\right|,q)$ and ${\theta }_2(x,q)={\theta }_1(\frac{1}{2}-\left|x\right|,q)$.

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