Pagina:Werkecarlf03gausrich.djvu/170

$\cos(n-1)\varphi .\Psi (-{\tfrac {1}{n}})=-{\tfrac {n}{n-1}}\cos(n-1)\varphi +\cos(n-1)\varphi .\log 2-{\tfrac {n}{2n-1}}\cos(2n-1)\varphi$
$\qquad \qquad +\cos(n-1)\varphi .\log {\tfrac {3}{2}}-$ etc.

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

atque per summationem

$\cos \varphi .\Psi {\tfrac {1-n}{n}}+\cos 2\varphi .\Psi {\tfrac {2-n}{n}}+\cos 3\varphi .\Psi {\tfrac {3-n}{n}}+$ etc. $+\cos(n-1)\varphi .\Psi (-{\tfrac {1}{n}})+\Psi 0$

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

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

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

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

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

Statuatur in hac aequatione deinceps $\varphi =\omega ,$ $\varphi =2\omega ,$ $\varphi =3\omega$ etc. usque ad $\varphi =(n-1)\omega ,$ 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

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

Quodsi iam perpenditur, esse

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

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

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

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