C349 Jacobian Theta Functions

Function subprograms RTHETA and DTHETA calculate the Jacobian theta functions

$\vartheta$0(x,q)	$=$	$1\; +\; 2\; \sum$n=1&inf;(-1)nqn2cos2n πx,
$\vartheta$1(x,q)	$=$	$2\; \sum$n=0&inf;(-1)nq( n+{12})2sin(2n+1) πx,
$\vartheta$2(x,q)	$=$	$2\; \sum$n=0&inf;q( n+{12})2cos(2n+1) πx,
$\vartheta$3(x,q)	$=$	$1\; +\; 2\; \sum$n=1&inf;qn2cos2n πx,
$\vartheta$4(x,q)	$=$	$\vartheta$0(x,q),

for real arguments x and . $\vartheta$₁(x+{12},1)

and $\vartheta$₂(x,1) are undefined if x is an integer; otherwise $\vartheta$_k(x,1)=0, k=1,2,3,4 .

Note that several conflicting definitions of these functions occur in the literature. In particular, the argument in the trigonometric terms is often defined to be x instead of $\pi x$ .

On CDC and Cray computers, the double-precision version DTHETA is not available.

Structure:

FUNCTION subprogram
User Entry Names: RTHETA, DTHETA
Files Referenced: Unit 6
External References: MTLMTR (N002), ABEND (Z035)

Usage:

In any arithmetic expression, RTHETA(K,X,Q) or DTHETA(K,X,Q) has the value $\vartheta$_K(X,Q) ,

where RTHETA is of type REAL, DTHETA is of type DOUBLE PRECISION, X and Q are of the same type as the function name, and K is of type INTEGER.

Method:

If $t\hspace{0.17em}(0\; \le t\; \le \{12\})$ differs from x or -x by an integer, it follows from the periodicity and symmetry properties of the functions that $\vartheta$₁(x,q)=±ϑ₁(t,q) and $\vartheta$₃(x,q)=ϑ₃(t,q) . In a region for which the approximation is sufficiently accurate, $\vartheta$₁ is set equal to the first $(n=0)$ term of the transformed series

$\vartheta$₁(t,q) = 2(λ/π)^1/2e^-λt2∑_n=0^&inf;(-1)ⁿe^-λ(n+{12})2sinh(2n+1)λt,

and $\vartheta$₃ is set equal to the first two (i.e. $n\; \le 1$ ) terms of

$\vartheta$₃(t,q) = (λ/π)^1/2e^-λt2( 1+2 ∑_n=1^&inf;e^-λn2cosh2nλt ),

where $\lambda =\; \pi 2/|lnq|$ . Otherwise the trigonometric series for $\vartheta$₁(t,q) and $\vartheta$₃(t,q) are used.

For all x, $\vartheta$₀ and $\vartheta$₂ are computed from $\vartheta$₀(x,q)=ϑ₃({12}-|x|,q) , $\vartheta$₂(x,q)=ϑ₁({12}-|x|,q) .

Restrictions:

1. $0\; \le Q\; \le 1$ .
2. $K\; =\; 0,1,2,3,4$ .
3. If $Q\; =\; 1$ and $K\; =\; 1$ , $X-\{12\}$ must not be an integer.
If $Q\; =\; 1$ and $K\; =\; 2$ , $X$ must not be an integer.

Error handling:

Error C349.1: Restriction 1 is not satisfied.
Error C349.2: Restriction 2 is not satisfied.
Error C349.3: Restriction 3 is not satisfied.
In all cases, the function value is set equal to zero, and a message is written on Unit 6, unless subroutine MTLSET (N002) has been called.

Accuracy:

For DTHETA (and for RTHETA on CDC and Cray computers), the error when Q is less than approximately 0.9 does not exceed two decimal digits in the last place. For larger values of Q (provided the computed result is non-zero), the error is at worst comparable in magnitude to the mathematical error which would be caused by one-bit rounding errors in the arguments X and Q.

On computers other than CDC and Cray, non-zero values of RTHETA have full machine accuracy.

Notes:

Successive references using the same value of Q are executed faster than those in which Q changes.

Many functional relations, including relations between the theta functions and the Jacobian elliptic functions, are given in Refs. 1--4.

References:

1. W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and theorems for the special functions of mathematical physics, Springer-Verlag Berlin (1966) 371--377.
2. F. Tölke, Praktische Funktionenlehre, Bd. II, Springer-Verlag Berlin (1966) 1--38.
3. P.F. Byrd and M.D. Friedman, Handbook of elliptic integrals for engineers and scientists, 2nd Edition, Springer-Verlag Berlin (1971) 315--320.
4. E.T. Whittaker and G.N. Watson, A course of modern analysis, 4th Edition, Cambridge University Press, Cambridge (1946) Chapter 21.

Integration, Minimization, Non-linear Fitting

D101