Pagina:Werkecarlf03gausrich.djvu/174

37.

Statuendo in formula 77, $z=e^{-u},$ patet, illam etiam ita exhiberi posse

	$\Psi \lambda =-\int \left({\frac {e^{-u}}{u}}-{\frac {e^{-u\lambda -u}}{1-e^{-u}}}\right)\operatorname {d} u,$ ab $u=\infty$ usque ad $u=0,$ i.e.
[78]	$\Psi \lambda =\int \left({\frac {e^{-u}}{u}}-{\frac {e^{-2u}}{e^{u}-1}}\right)\operatorname {d} u,$ ab $u=0$ usque ad $u=\infty$

(Perinde valor ipsius $\Pi \lambda$ in art. 28 allatus, mutatur statuendo $e^{-y}=0,$ in sequentem

$\Pi \lambda =\int \left(\log {\frac {1}{v}}\right)^{\lambda }dv,$ a $v=0$ usque ad $v=1$)

Porro patet e formula 77, esse

[79]	$\Psi \lambda -\Psi \mu =\int {\frac {z^{\mu }-z^{\lambda }}{1-z}}\operatorname {d} z,$ a $z=0$ usque ad $z=1$

ubi praeter $\lambda +1$ etiam $\mu +1$ debet esse quantitas positiva.

Statuendo in eadem formula 77, $z=u^{\alpha },$ designante $\alpha$ quantitatem positivam, fit

$\Psi \lambda =\int \left(-{\frac {u^{\alpha -1}}{\log u}}-{\frac {au^{\alpha \lambda +\alpha -1}}{1-u^{\alpha }}}\right)\operatorname {d} u,$ ab $u=0$ usque ad $u=1$

et quum perinde statui possit, pro valore positivo ipsius $\beta ,$

$\Psi \lambda =\int \left(-{\frac {u^{\beta -1}}{\log u}}-{\frac {\beta u^{\beta \lambda +\beta -1}}{1-u^{\beta }}}\right)\operatorname {d} u$

patet, fieri

$0=\int \left({\frac {u^{\alpha -1}-u^{\beta -1}}{\log u}}+{\frac {\alpha u^{\alpha \lambda +\alpha -1}}{1-u^{\alpha }}}-{\frac {\beta u^{\beta \lambda +\beta -1}}{1-u^{\beta }}}\right)\operatorname {d} u$

sive

$\int {\frac {u^{\alpha -1}-u^{\beta -1}}{\log u}}\operatorname {d} u=\int \left({\frac {\beta u^{\beta \lambda +\beta -1}}{1-u^{\beta }}}-{\frac {\alpha u^{\alpha \lambda +\alpha -1}}{1-u^{\alpha }}}\right)\operatorname {d} u$

integralibus semper ab $u=0$ usque ad $u=1$ extensis. Sed ponendo $\lambda =0,$ integrale posterius indefinite assignari potest; est scilicet $=\log {\tfrac {1-u^{\alpha }}{1-u^{\beta }}},$ si evanescere debet pro $u=0;$ quare quum pro $u=1$ statuere oporteat ${\tfrac {1-u^{\alpha }}{1-u^{\beta }}}={\tfrac {\alpha }{\beta }},$ erit integrale $\log {\tfrac {\alpha }{\beta }}=\int {\frac {u^{\alpha -1}-u^{\beta -1}}{\log u}}\operatorname {d} u,$ ab $u=0$ usque ad $u=1,$ quod theorema olim ab ill. Euler per alias methodos erutum est.

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