Pagina:Werkecarlf03gausrich.djvu/164

[57]	${\frac {n^{nz}\Pi z.\Pi (z-{\frac {1}{n}}).\Pi (z-{\frac {2}{n}})\ldots \ldots \Pi (z-{\frac {n-1}{n}})}{\Pi nz}}={\frac {(2\pi )^{{\frac {1}{2}}(n-1)}}{\sqrt {n}}}$

27.

Integrale $\int x^{\lambda -1}(1-x^{\mu })^{\nu }\operatorname {d} x,$ ita acceptum, ut evanescat pro $x=0,$ exprimitur per seriem sequentem, siquidem $\lambda ,$ $\mu$ sunt quantitates positivae:

${\frac {x^{\lambda }}{\lambda }}-{\frac {\nu x^{\mu +\lambda }}{\mu +\lambda }}+{\frac {\nu (\nu -1)x^{2\mu +\lambda }}{1.2.(2\mu +\lambda )}}-{\text{etc.}}={\frac {x^{\lambda }}{\lambda }}F\left(-\nu ,{\frac {\lambda }{\mu }},{\frac {\lambda }{\mu }}+1,x^{\mu }\right)$

Hinc ipsius valor pro $x=1$ erit

$={\frac {\Pi {\frac {\lambda }{\mu }}.\Pi \nu }{\lambda \Pi ({\frac {\lambda }{\mu }}+v)}}$

Ex hoc theoremate omnes relationes, quas ill. Euler olim multo labore evolvit, sponte demanant. Ita e.g. statuendo

$\int {\frac {\operatorname {d} x}{\sqrt {(1-x^{4})}}}=A,\quad \int {\frac {xx\operatorname {d} x}{\sqrt {(1-x^{4})}}}=B$

erit $A={\tfrac {\Pi {\frac {1}{4}}.\Pi (-{\frac {1}{2}})}{\Pi (-{\frac {1}{4}})}},$ $B={\tfrac {\Pi {\frac {3}{4}}.\Pi (-{\frac {1}{2}})}{3\Pi {\frac {1}{4}}}}={\tfrac {\Pi (-{\frac {1}{4}}).\Pi (-{\frac {1}{2}})}{4\Pi {\frac {1}{4}}}},$ adeoque $AB={\tfrac {1}{4}}\pi .$ Simul hinc sequitur, quoniam $\Pi {\tfrac {1}{4}}.\Pi (-{\tfrac {1}{4}})={\tfrac {{\frac {1}{4}}\pi }{\sin {\frac {1}{4}}\pi }}={\frac {\pi }{\sqrt {8}}},$

$\Pi {\tfrac {1}{4}}={\sqrt[{4}]{{\tfrac {1}{8}}\pi AA}}={\sqrt[{4}]{\frac {\pi ^{3}}{128BB}}},\quad \Pi (-{\tfrac {1}{4}})={\sqrt[{4}]{\frac {\pi ^{3}}{8AA}}}={\sqrt[{4}]{2\pi BB}}$

Valor numericus ipsius $A,$ computante Stirling, habetur $=1{,}3110287771\,4605987,$ valor ipsius $B,$ secundum eundem auctorem, $=0{,}5990701173\,6779611,$ ex nostro calculo, artificio peculiari innixo, $=0{,}5990701173\,6779610372.$

Generaliter facile ostendi potest, valorem functionis $\Pi {z},$ si $z$ sit quantitas rationalis $={\tfrac {m}{\mu }},$ denotantibus $m,$ $\mu$ integros, ex $\mu -1$ valoribus determinatis talium integralium pro $x=1$ deduci posse, et quidem permultis modis diversis. Accipiendo enim pro $\lambda$ numerum integrum atque pro $\nu$ fractionem, cuius denominator $=\mu ,$ valor illius integralis semper reducitur ad tres $\Pi z,$ ubi $z$ est fractio cum denominatore $=\mu ;$ quodvis vero huiusmodi $\Pi z$ vel ad $\Pi (-{\tfrac {1}{\mu }}),$ vel ad $\Pi (-{\tfrac {2}{\mu }}),$ vel ad $\Pi (-{\tfrac {3}{\mu }})$ etc. vel ad $\Pi (-{\tfrac {\mu -1}{\mu }})$ reduci potest per formulam 45, siquidem $z$ revera est fractio; si enim $z$ est integer, $\Pi {z}$ per se constat. Ex illis vero integralium valoribus, generaliter loquendo, quodvis $\Pi (-{\tfrac {m}{\mu }}),$