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Evaluation of the Multiple Sine Function...

DreamWeaver

Well-known member
Sep 16, 2013
337
Part 1:


Define the Multiple Sine Function by

\(\displaystyle \mathcal{S}_m(x)=\text{exp}\left(\frac{x^{m-1}}{m-1}\right)
\prod_{k=1}^{\infty}\left(\mathcal{P}_m\left(\frac{x}{k}\right)\mathcal{P}_m\left(-\frac{x}{k}\right)^p\right)^q\)

Where \(\displaystyle p=(-1)^{m-1}\,\) and \(\displaystyle q=k^{m-1}\,\) (these exponents weren't showing up so well in the above, hence the abbreviations). The function \(\displaystyle \mathcal{P}_n(z)\,\) is defined by:

\(\displaystyle \mathcal{P}_n(z)=(1-z)\,\text{exp}\left(z+\frac{z^2}{2}+\frac{z^3}{3}+\, \cdots \, +\frac{z^n}{n}\right)\)





The challenge is this: for \(\displaystyle 0\le\theta <\pi\,\) and \(\displaystyle m\ge 2\,\), prove that:



\(\displaystyle \int_0^{\theta}x^{m-2}\log(\sin x)\,dx=\frac{\theta^{m-1}}{(m-1)}\log(\sin \theta)-\frac{\pi^{m-1}}{(m-1)}\log\mathcal{S}_m\left(\frac{\theta}{\pi}\right)\)