Pagina:Werkecarlf03gausrich.djvu/170

$\cos (n-1)\phi .\mathrm{\Psi }(-\frac{1}{n})=-\frac{n}{n-1}\cos (n-1)\phi +\cos (n-1)\phi .\log 2-\frac{n}{2n-1}\cos (2n-1)\phi$
$+\cos (n-1)\phi .\log \frac{3}{2}-$ etc.

$\phantom{\cos (n-1)\phi -}\mathrm{\Psi }0=-\frac{n}{n}\cos n\phi +\log 2-\frac{n}{2n}\cos 2n\phi +\log \frac{3}{2}-$ etc.

atque per summationem

$\cos \phi .\mathrm{\Psi }\frac{1-n}{n}+\cos 2\phi .\mathrm{\Psi }\frac{2-n}{n}+\cos 3\phi .\mathrm{\Psi }\frac{3-n}{n}+$ etc. $+\cos (n-1)\phi .\mathrm{\Psi }(-\frac{1}{n})+\mathrm{\Psi }0$

$=-n(\cos \phi +\frac{1}{2}\cos 2\phi +\frac{1}{3}\cos 3\phi +\frac{1}{4}\cos 4\phi +$ etc. in infin.$)$

Sed habetur generaliter, pro valore ipsius $x$ unitate non maiori,

$\log (1-2x\cos \phi +xx)=-2(x\cos \phi +\frac{1}{2}xx\cos 2\phi +\frac{1}{3}{x}^3\cos 3\phi +\text{etc.})$

quae quidem series facile sequitur ex evolutione $\log (1-rx)+\log (1-\frac{x}{r}),$ denotante $r$ quantitatem $\cos \phi +\sqrt{-1}.\sin \phi .$ Hinc fit aequatio praecedens

[72]	$\cos \phi .\mathrm{\Psi }\frac{1-n}{n}+\cos 2\phi .\mathrm{\Psi }\frac{2-n}{n}+\cos 3\phi .\mathrm{\Psi }\frac{3-n}{n}+\text{etc.}+\cos (n-1)\phi .\mathrm{\Psi }(-\frac{1}{n})$
	$=-\mathrm{\Psi }0+\frac{1}{2}n\log (2-2\cos \phi )$

Statuatur in hac aequatione deinceps $\phi =\omega ,$ $\phi =2\omega ,$ $\phi =3\omega$ etc. usque ad $\phi =(n-1)\omega ,$ multiplicentur singulae hae aequationes ordine suo per $\cos m\omega ,$ $\cos 2m\omega ,$ $\cos 3m\omega$ etc. usque ad $\cos (n-1)m\omega ,$ productorumque aggregato adiiciatur aequatio 71

$\mathrm{\Psi }\frac{1-n}{n}+\mathrm{\Psi }\frac{2-n}{n}+\mathrm{\Psi }\frac{3-n}{n}+\text{etc.}+\mathrm{\Psi }(-\frac{1}{n})=(n-1)\mathrm{\Psi }0-n\log n$

Quodsi iam perpenditur, esse

$\begin{array}{cc}1+\cos m\omega .\cos k\omega +\cos 2m\omega .\cos 2k\omega +\cos 3m\omega .\cos 3k\omega & \\ +\text{ etc.}+\cos (n-1)m\omega .\cos (n-1)k\omega & =0\end{array}$

denotante $k$ aliquem numerorum $1,$ $2,$ $3\dots (n-1)$ exceptis his duobus $m$ atque $n-m,$ pro quibus summa illa fit $=\frac{1}{2}n,$ patebit, ex summatione illarum aequationum prodire, post divisionem per $\frac{n}{2},$

[73]	$\mathrm{\Psi }(-\frac{m}{n})+\mathrm{\Psi }(-\frac{n-m}{m})=$ $2\mathrm{\Psi }0-2\log n+\cos m\omega .\log (2-2\cos \omega )+\cos 2m\omega .\log (2-2\cos 2\omega )$ $+\cos 3m\omega .\log (2-2\cos 3\omega )+\text{etc.}+\cos (n-1)m\omega .\log (2-2\cos (n-1)\omega )$

Manifesto terminus ultimus huius aequationis fit $=\cos m\omega .\log (2-2\cos \omega ),$ pen-