NAG Library Function Document

nag_jacobian_theta (s21ccc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_jacobian_theta (s21ccc) returns the value of one of the Jacobian theta functions

θ_{0} (x, q)

θ_{1} (x, q)

θ_{2} (x, q)

θ_{3} (x, q)

θ_{4} (x, q)

for a real argument

x

and non-negative

q \leq 1

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_jacobian_theta (Integer k, double x, double q, NagError *fail)

3 Description

nag_jacobian_theta (s21ccc) evaluates an approximation to the Jacobian theta functions

θ_{0} (x, q)

θ_{1} (x, q)

θ_{2} (x, q)

θ_{3} (x, q)

and

θ_{4} (x, q)

given by

\begin{aligned} θ_{0} (x, q) & = 1 + 2 \sum_{n = 1}^{\infty} {(- 1)}^{n} q^{n^{2}} \cos (2 n π x), \\ θ_{1} (x, q) & = 2 \sum_{n = 0}^{\infty} {(- 1)}^{n} q^{{(n + \frac{1}{2})}^{2}} \sin \{(2 n + 1) π x\}, \\ θ_{2} (x, q) & = 2 \sum_{n = 0}^{\infty} q^{{(n + \frac{1}{2})}^{2}} \cos \{(2 n + 1) π x\}, \\ θ_{3} (x, q) & = 1 + 2 \sum_{n = 1}^{\infty} q^{n^{2}} \cos (2 n π x), \\ θ_{4} (x, q) & = θ_{0} (x, q), \end{aligned}

where

x

and

q

(the nome) are real with

0 \leq q \leq 1

. Note that

θ_{1} (x - \frac{1}{2}, 1)

is undefined if

(x - \frac{1}{2})

is an integer, as is

θ_{2} (x, 1)

x

is an integer; otherwise,

θ_{i} (x, 1) = 0

, for

i = 0, 1, \dots, 4

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., Abramowitz and Stegun (1972), 16.27) 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_jacobian_theta (s21ccc) 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)

4 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

5 Arguments

1: k – IntegerInput

On entry: 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 – doubleInput

On entry: the argument

x

of the function.

Constraints:

x must not be an integer when $q = 1.0$ and $k = 2$ ;
$(x - 0.5)$ must not be an integer when $q = 1.0$ and $k = 1$ . .

3: q – doubleInput

On entry: the argument

q

of the function.

Constraint:

0.0 \leq q \leq 1.0

4: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_INFINITE: The evaluation has been abandoned because the function value is infinite.
NE_INT: On entry, $k = 〈value〉$ .
Constraint: $0 \leq k \leq 4$ .
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.
NE_REAL: On entry, $q = 〈value〉$ .
Constraint: $0.0 \leq q \leq 1.0$ .; On entry, $x = 〈value〉$ .
Constraint: $(x - 0.5)$ must not be an integer when $q = 1.0$ and $k = 1$ .; On entry, $x = 〈value〉$ .
Constraint: x must not be an integer when $q = 1.0$ and $k = 2$ .

7 Accuracy

In principle nag_jacobian_theta (s21ccc) is capable of achieving full relative precision in the computed values. However, the accuracy obtainable in practice depends on the accuracy of the C standard library elementary functions such as sin and cos.

8 Further Comments

None.

9 Example

The example program evaluates

θ_{2} (x, q)

x = 0.7

when

q = 0.4

, and prints the results.

NAG Library Function Documentnag_jacobian_theta (s21ccc)