Pagina:Werkecarlf03gausrich.djvu/169

32.

Formula 55 nobis suppeditat $\log \Pi (-z)+\log \Pi (z-1)=\log \pi -\log \sin z\pi ,$ unde fit per differentiationem

[68]	$\Psi (-z)-\Psi (z-1)=\pi \operatorname {cotang} z\pi$

Et quum e definitione functionis $\Psi$ generaliter habeatur

[69]	$\Psi x-\Psi y=-{\frac {1}{x+1}}+{\frac {1}{y+1}}-{\frac {1}{x+2}}+{\frac {1}{y+2}}-{\frac {1}{x+3}}+$ etc.

oritur series nota

$\pi {\text{ cotang }}z\pi ={\frac {1}{z}}-{\frac {1}{1-z}}+{\frac {1}{1+z}}-{\frac {1}{2-z}}+{\frac {1}{2+z}}-{\frac {1}{3-z}}+$ etc.

Simili modo e differentiatione formulae 57 prodit

[70]	$\Psi z+\Psi (z-{\tfrac {1}{n}})+\Psi (z-{\tfrac {2}{n}})+$ etc. $+\Psi (z-{\tfrac {n-1}{n}})=n\Psi nz-n\log n$

adeoque statuendo $z=0$

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

Ita e.g. habetur

$\Psi (-{\tfrac {1}{2}})=\Psi 0-2\log 2=-1,9635100260\;\;2142347944\;\;099,$ unde porro

$\qquad \qquad \Psi {\tfrac {1}{2}}=+0,0364899739\;\;7857652055\;\;901.$

33.

Sicuti in art. praec. $\Psi (-{\tfrac {1}{2}})$ ad $\Psi 0$ et logarithmum reduximus, ita generaliter $\Psi (-{\tfrac {m}{n}}),$ designantibus $m,n$ integros, quorum minor $m,$ ad $\Psi 0$ et logarithmos reducemus. Statuamus ${\tfrac {2\pi }{n}}=\omega ,$ sitque $\varphi$ alicui angulorum $\omega ,$ $2\omega ,$ $3\omega \ldots (n-1)\omega$ aequalis; unde $1=\cos n\varphi =\cos 2n\varphi =\cos 3n\varphi$ etc., $\cos \varphi =\cos(n+1)\varphi =\cos(n+2)\varphi$ etc., $\cos 2\varphi =\cos(n+2)\varphi$ etc., nec non $\cos \varphi +\cos 2\varphi +\cos 3\varphi +$ etc. $+\cos(n-1)\varphi +1=0.$ Habemus itaque

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

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

etc. usque ad