Pagina:Werkecarlf03gausrich.djvu/163

E combinatione formulae 54 cum definitione functionis $\Pi$ sequitur, ${\tfrac {z\pi }{\sin z\pi }}$ esse limitem producti infiniti

${\frac {(1\;.\;2\;.\;3\;.\;4\;\ldots \;k)^{2}}{(1-zz)(4-zz)(9-zz)\ldots (kk-zz)}}$

crescente $k$ in infinitum, adeoque

$\sin z\pi =z\pi \left(1-zz\right)\left(1-{\tfrac {zz}{4}}\right)\left(1-{\tfrac {zz}{9}}\right)$ etc. in inf.

similique modo ex 56 deducitur

$\cos z\pi =\left(1-4zz\right)\left(1-{\tfrac {4zz}{9}}\right)\left(1-{\tfrac {4zz}{25}}\right)$ etc. in inf.

formulae notissimae, quae ab analystis per methodos prorsus diversas erui solent.

26.

Designante $n$ numerum integrum, valor expressionis

${\frac {n^{nz}\Pi (k,z).\Pi (k,z-{\frac {1}{n}}).\Pi (k,z-{\frac {2}{n}})\ldots \ldots \Pi (k,z-{\frac {n-1}{n}})}{\Pi (nk,nz)}}$

rite collectus invenitur

$={\frac {(1\;.\;2\;.\;3\;\ldots \;k)^{n}n^{nk}}{1\;.\;2\;.\;3\;\ldots \;nk.k^{{\frac {1}{2}}(n-1)}}}$

adeoque a $z$ est independens, sive idem manebit, quicunque valor ipsi $z$ tribuatur. Exhiberi poterit itaque, quoniam $\Pi (k,0)=\Pi (nk,0)=1,$ per productum

$\Pi (k,-{\tfrac {1}{n}}).\Pi (k,-{\tfrac {2}{n}}).\Pi (k,-{\tfrac {3}{n}})\ldots \Pi (k,-{\tfrac {n-1}{n}})$

Crescente igitur $k$ in infinitum, nanciscimur

${\frac {n^{nz}\Pi z.\Pi (z-{\tfrac {1}{n}}).\Pi (z-{\tfrac {2}{n}})\ldots \Pi (z-{\tfrac {n-1}{n}})}{\Pi nz}}=\Pi (-{\tfrac {1}{n}}).\Pi (-{\tfrac {2}{n}}).\Pi (-{\tfrac {3}{n}})\ldots \Pi (-{\tfrac {n-1}{n}})$

Productum ad dextram, in se ipsum ordine factorum inverso multiplicatum, producit, per form. 55,

${\frac {\pi }{\sin {\frac {1}{n}}\pi }}.{\frac {\pi }{\sin {\frac {2}{n}}\pi }}.{\frac {\pi }{\sin {\frac {3}{n}}\pi }}\cdots {\frac {\pi }{\sin {\frac {n-1}{n}}\pi }}={\frac {(2\pi )^{n-1}}{n}}$

Unde habemus theorema elegans