Pagina:Werkecarlf03gausrich.djvu/139

DISQUISITIONES GENERALES

CIRCA SERIEM INFINITAM

$1+{\frac {\alpha \beta }{1.\gamma }}x+{\frac {\alpha (\alpha +1)\beta (\beta +1)}{1.2.\gamma (\gamma +1)}}xx+{\frac {\alpha (\alpha +1)(\alpha +2)\beta (\beta +1)(\beta +2)}{1.2.3.\gamma (\gamma +1)(\gamma +2)}}x^{3}+{\text{ etc.}}$

PARS PRIOR.

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INTRODUCTIO.

1.

Series, quam in hac commentatione perscrutari suscipimus, tamquam functio quatuor quantitatum $\alpha ,$ $\beta ,$ $\gamma ,$ $x$ spectari potest, quas ipsius elementa vocabimus, ordine suo elementum primum $\alpha ,$ secundum $\beta ,$ tertium $\gamma ,$ quartum $x$ distinguentes. Manifesto elementum primum cum secundo permutare licet: quodsi itaque brevitatis caussa seriem nostram hoc signo $F(\alpha ,\beta ,\gamma ,x)$ denotamus, habebimus $F(\alpha ,\beta ,\gamma ,x)=F(\beta ,\alpha ,\gamma ,x).$

2.

Tribuendo elementis $\alpha ,$ $\beta ,$ $\gamma$ valores determinatus, series nostra in functionem unicae variabilis $x$ transit, quae manifesto post terminum $1-\alpha ^{tum}$ vel $1-\beta ^{tum}$ abrumpitur, si $\alpha -1$ vel $\beta -1$ est numerus integer negativus, in casibus reliquis vero in infinitum excurrit. In casu priori series exhibet functionem algebraicam rationalem, in posteriori autem plerumque functionem transscendentem. Elementum tertium $\gamma$ debet esse neque numerus negativus integer neque $=0,$ ne ad terminos infinite magnos delabamur.