Pagina:Werkecarlf03gausrich.djvu/167

30.

E formula 38 sequitur

${\frac {\Pi (k,z+\omega )}{\Pi (k,z)}}={\frac {z+1}{z+1+\omega }}.{\frac {z+2}{z+2+\omega }}.{\frac {z+3}{z+3+\omega }}\ldots .{\frac {z+k}{z+k+\omega }}.k^{\omega }$

unde sumtis logarithmis, in series infinitas evolutis, prodit

[60]

${\begin{aligned}\log \Pi (k,z+\omega )=&\log \Pi (k,z)\\&+\omega \left(\log k-{\tfrac {1}{z+1}}-{\tfrac {1}{z+2}}-{\tfrac {1}{z+3}}-{\text{etc. }}-{\tfrac {1}{z+k}}\right)\\&+{\tfrac {1}{2}}\omega \omega \left({\tfrac {1}{(z+1)^{2}}}+{\tfrac {1}{(z+2)^{2}}}+{\tfrac {1}{(z+3)^{3}}}+{\text{etc.}}+{\tfrac {1}{(z+k)^{2}}}\right)\\&-{\tfrac {1}{3}}\omega ^{3}\left({\tfrac {1}{(z+1)^{3}}}+{\tfrac {1}{(z+2)^{3}}}+{\tfrac {1}{(z+3)^{3}}}+{\text{etc.}}+{\tfrac {1}{(z+k)^{3}}}\right)\\&+{\text{ etc. in inf.}}\end{aligned}}$

Series, hic in $\omega$ multiplicata, quae, si magis placet, ita etiam exhiberi potest,

$-{\tfrac {1}{z+1}}+\log 2-{\tfrac {1}{z+2}}+\log {\tfrac {3}{2}}-{\tfrac {1}{z+3}}+\log {\tfrac {4}{3}}-{\tfrac {1}{z+4}}+\log {\tfrac {5}{4}}-{\text{etc.}}+\log {\tfrac {k}{k-1}}-{\tfrac {1}{z+k}}$

e terminorum multitudine finita constat, crescente autem $k$ in infinitum, ad limitem certum converget, qui novam functionum transscendentium speciem nobis sistit, in posterum per $\Psi z$ denotandam.

Designando porro summas serierum sequentium, in infinitum extensarum,

${\begin{aligned}&{\tfrac {1}{(z+1)^{2}}}+{\tfrac {1}{(z+2)^{2}}}+{\tfrac {1}{(z+3)^{2}}}+{\text{etc. }}\\&{\tfrac {1}{(z+1)^{3}}}+{\tfrac {1}{(z+2)^{3}}}+{\tfrac {1}{(z+3)^{3}}}+{\text{etc. }}\\&{\tfrac {1}{(z+1)^{4}}}+{\tfrac {1}{(z+2)^{4}}}+{\tfrac {1}{(z+3)^{4}}}+{\text{etc. }}\\&{\text{ etc. }}\end{aligned}}$

resp. per ${P},$ ${Q},$ ${R}$ etc. (pro quibus signa functionalia introducere minus necessarium videtur), habebimus

[61]	$\log \Pi (z+\omega )=\log \Pi z+\omega \Psi z+{\tfrac {1}{2}}\omega \omega P-{\tfrac {1}{3}}\omega ^{3}Q+{\tfrac {1}{4}}\omega ^{4}R-{\text{etc.}}$

Manifesto functio $\Psi z$ erit functio derivata prima functionis $\log \Pi z,$ adeoque

[62]	${\frac {\operatorname {d} \Pi z}{\operatorname {d} z}}=\Pi z.\Psi z$

Perinde erit $P={\frac {\operatorname {d} \Psi z}{\operatorname {d} z}},Q=-{\frac {\mathrm {dd} \Psi _{z}}{2\operatorname {d} z^{2}}},R=+{\frac {\operatorname {d} ^{3}\Psi z}{2.3\operatorname {d} z^{3}}}$ etc.