# clausen

clausen, an Octave code which evaluates a Chebyshev interpolant to the Clausen function Cl2(x).

Cl2(x) = - integral ( 0 < s < x ) log ( abs ( 2 sin ( x / 2 ) ) ) dx

The Clausen function is periodic, with period 2 pi.

### Languages:

clausen is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

### Related Data and Programs:

chebyshev, an Octave code which computes the Chebyshev interpolant/approximant to a given function over an interval.

chebyshev_interp_1d, an Octave code which determines the combination of Chebyshev polynomials which interpolates a set of data, so that p(x(i)) = y(i).

chebyshev_polynomial, an Octave code which considers the Chebyshev polynomials T(i,x), U(i,x), V(i,x) and W(i,x). Functions are provided to evaluate the polynomials, determine their zeros, produce their polynomial coefficients, produce related quadrature rules, project other functions onto these polynomial bases, and integrate double and triple products of the polynomials.

chebyshev_series, an Octave code which can evaluate a Chebyshev series approximating a function f(x), while efficiently computing one, two or three derivatives of the series, which approximate f'(x), f''(x), and f'''(x), by Manfred Zimmer.

fn, an Octave code which approximates elementary and special functions using Chebyshev polynomials; functions include Airy, Bessel I, J, K and Y, beta, confluent hypergeometric, error, gamma, log gamma, Pochhammer, Spence; integrals include hyperbolic cosine, cosine, Dawson, exponential, logarithmic, hyperbolic sine, sine; by Wayne Fullerton.

polpak, an Octave code which evaluates a variety of mathematical functions, including Chebyshev, Gegenbauer, Hermite, Jacobi, Laguerre, Legendre polynomials, and the Collatz sequence.

test_values, an Octave code which supplies test values of various mathematical functions.

toms446, an Octave code which manipulates Chebyshev series for interpolation and approximation; this is ACM TOMS algorithm 446, by Roger Broucke.

### Reference:

1. Kurt Koelbig,
Chebyshev coefficients for the Clausen function Cl2(x),
Journal of Computational and Applied Mathematics,
Volume 64, Number 3, 1995, pages 295-297.

### Source Code:

• clausen.m, evaluates the Clausen function using a Chebyshev series.
• clausen_values.m, returns a few selected values of the Clausen function.
• r8_csevl.m, evaluates a Chebyshev series.
• r8_wrap.m, "wraps" a real value periodically so it lies in a given interval.

Last revised on 13 December 2018.