Pagina:Werkecarlf03gausrich.djvu/171

ultimus $=\cos 2m\omega .\log(2-2\cos 2\omega )$ etc., ita ut bini termini semper sint aequales, excepto, si $n$ est par, termino singulari $\cos {\tfrac {n}{2}}.m\omega \log(2-2\cos {\tfrac {n}{2}}\omega ),$ qui fit $=+2\log 2$ pro $m$ pari, vel $=-2\log 2$ pro $m$ impari. Combinando iam cum aequatione 73 hanc

$\Psi (-{\tfrac {m}{n}})-\Psi (-{\tfrac {n-m}{n}})=\pi \operatorname {cotang} {\tfrac {m}{n}}\pi$

habemus, pro valore impari ipsius $n,$ siquidem $m$ est integer positivus minor quam $n$

[74]

${\begin{aligned}\Psi (-{\tfrac {m}{n}})=&\Psi 0+{\tfrac {1}{2}}\pi \operatorname {cotang} {\tfrac {m\pi }{n}}-\log n+\cos {\tfrac {2m\pi }{n}}.\log(2-2\cos {\tfrac {2\pi }{n}})\\&+\cos {\tfrac {4m\pi }{n}}.\log(2-2\cos {\tfrac {4\pi }{n}})+\cos {\tfrac {6m\pi }{n}}.\log(2-2\cos {\tfrac {6\pi }{n}})+{\text{etc. }}\\&+\cos {\tfrac {(n-1)m\pi }{n}}.\log(2-2\cos {\tfrac {(n-1)\pi }{n}})\end{aligned}}$

Pro valore pari ipsius $n$ autem

[75]

${\begin{aligned}\Psi (-{\tfrac {m}{n}})=&\Psi 0+{\tfrac {1}{2}}\pi \operatorname {cotang} {\tfrac {m\pi }{n}}-\log n+\cos {\tfrac {2m\pi }{n}}\log(2-2\cos {\tfrac {2\pi }{n}})\\&+\cos {\tfrac {4m\pi }{n}}\log(2-2\cos {\tfrac {4\pi }{n}})+{\text{etc.}}+\cos {\tfrac {(n-2)m\pi }{n}}\log(2-2\cos {\tfrac {(n-2)\pi }{n}})\\&\pm \log 2\end{aligned}}$

ubi signum superius valet pro $m$ pari, inferius pro impari. Ita e.g. invenitur

${\begin{array}{ll}\Psi (-{\tfrac {1}{4}})=\Psi 0+{\tfrac {1}{2}}\pi -3\log 2,&\Psi (-{\tfrac {3}{4}})=\Psi 0-{\tfrac {1}{2}}\pi -3\log 2\\\Psi (-{\tfrac {1}{3}})=\Psi 0+{\tfrac {1}{2}}\pi {\sqrt {\tfrac {1}{3}}}-{\tfrac {3}{2}}\log 3,&\Psi (-{\tfrac {2}{3}})=\Psi 0-{\tfrac {1}{2}}\pi {\sqrt {\tfrac {1}{3}}}-{\tfrac {3}{2}}\log 3\end{array}}$

Ceterum combinatis his aequationibus cum aequatione 64 sponte patet, $\Psi z$ generaliter pro quovis valore rationali ipsius $z,$ positivo seu negativo per $\Psi {0}$ atque logarithmos determinari posse, quod theorema sane maxime est memorabile.

34.

Quum, per art. 28, $\Pi \lambda$ sit valor integralis $\int y^{\lambda }e^{-y}\operatorname {d} y,$ ab $y=0$ usque ad $y=\infty ,$ siquidem $\lambda +1$ est quantitas positiva, fit differentiando secundum $\lambda$

${\frac {\operatorname {d} \Pi \lambda }{\operatorname {d} \lambda }}={\frac {\operatorname {d} \int y^{\lambda }e^{-y}\operatorname {d} y}{\operatorname {d} \lambda }}=\int y^{\lambda }e^{-y}\log y\operatorname {d} y$

sive

[76]	$\Pi \lambda .\Psi \lambda =\int y^{\lambda }e^{-y}\log y.\operatorname {d} y,$ ab $y=0$ usque ad $y=\infty$