s21cc 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 < 1

Syntax

C#
public static double s21cc( int k, double x, double q, out int ifail )

Visual Basic
Public Shared Function s21cc ( _ k As Integer, _ x As Double, _ q As Double, _ <OutAttribute> ByRef ifail As Integer _ ) As Double

Visual C++
public: static double s21cc( int k, double x, double q, [OutAttribute] int% ifail )

F#
static member s21cc : k : int * x : float * q : float * ifail : int byref -> float

Parameters

k: Type: System..::..Int32
On entry: denotes the function $θ_{k} (x, q)$ to be evaluated. Note that $k = 4$ is equivalent to $k = 0$ .

Constraint: $0 \leq k \leq 4$ .

x: Type: System..::..Double
On entry: the argument $x$ of the function.

q: Type: System..::..Double
On entry: the argument $q$ of the function.

Constraint: $0.0 \leq q < 1.0$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

s21cc 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 < 1

Description

s21cc 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{array}{lcl} 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{array}

where

x

and

q

(the nome) are real with

0 \leq q < 1

These functions are important in practice because every one of the Jacobian elliptic functions (see 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.

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

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$

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

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

$ifail = -9000$: An error occured, see message report.

Accuracy

In principle the method 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.