Formulae concerning the computation of the Clausen integral \(Cl_ 2(\theta)\) (Q802062)

The same problem of expressing the Clausen integral \(Cl_ 2(\theta)\) as in the paper reviewed above is given from a different point of view. Then, using an appropriate Fourier series expansion, the author gives a general formula for \(Cl_ 2(p\pi /q),\) where p and q represent positive integers satisfying \(0<p<2q\), \(q=1,2,...\), in terms of the derivative of the di-gamma function and the sine function. Other variants of this formula are examined. The formulae expressing \(Cl_ 2(2\theta)\) and, more generally, \(Cl_ 2(m\theta)\) \((m=2,3,...)\) as linear combinations of terms of the form \(Cl_ 2(\theta +\alpha)\) (\(\alpha\) const.) are obtained. The various results are illustrated by means of typical examples of practical application.