C326 Clausen Function

Function subprograms RCLAUS and DCLAUS calculate the Clausen function

$Cl$₂(x) = - ∫₀^xln|2 sin{t2}| dt = ∑_k=1^&inf;{sinkxk²}

for real arguments x.

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

Structure:

FUNCTION subprograms
User Entry Names: RCLAUS, DCLAUS

Usage:

In any arithmetic expression, RCLAUS(X) or DCLAUS(X) has the value $Cl$₂(X) ,

where RCLAUS is of type REAL, DCLAUS is of type DOUBLE PRECISION, and X has the same type as the function name.

Method:

For $0\; \le x\; \le \pi$ , the function is approximated by truncated Chebyshev series. For x outside this range, the relations $Cl$₂(π+x)=-Cl₂(π-x) and $Cl$₂(2nπ±x)=±Cl₂(x) are used.

Accuracy:

RCLAUS (except on CDC and Cray computers) has full single-precision accuracy in the interval $0\; \le x\; \le 2\pi$ . For most values of the argument $X\in [0,2\pi ]$ , DCLAUS (and RCLAUS on CDC and Cray computers) has an accuracy of approximately one significant digit less than the machine precision. Accuracy is lost near the zero of $Cl$₂(x) at $x=\pi$

and for large values of |x|.
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C327