Pagina:Werkecarlf03gausrich.djvu/142

6.

Functiones praecedentes sunt algebraicae atque transscendentes a logarithmis circuloque pendentes. Neutiquam vero harum caussa disquisitionem nostram generalem suscipimus, sed potius in gratiam theoriae functionum transscendentium altiorum promovendae, quarum genus amplissimum series nostra complectitur. Huc, inter infinita alia, pertinent coëfficientes ex evolutione functionis $(aa+bb-2ab\cos \varphi )^{-n}$ in seriem secundum cosinus angulorum $\varphi ,2\varphi ,3\varphi$ etc. progredientem orti, de quibus in specie alia occasione fusius agemus. Ad formam seriei nostrae autem illi coëfficientes pluribus modis reduci possunt. Scilicet statuendo

$(aa+bb-2ab\cos \varphi )^{-n}=\Omega =A+2A^{\prime }\cos \varphi +2A^{\prime \prime }\cos 2\varphi +2A^{\prime \prime \prime }\cos 3\varphi +$ etc.

habemus primo

${\begin{alignedat}{2}&A&&=a^{-2n}F(n,n,1,{\tfrac {bb}{aa}})\\&A^{\prime }&&=na^{-2n-1}bF(n,n+1,2,{\tfrac {bb}{aa}})\\&A^{\prime \prime }&&={\tfrac {n(n+1)}{1.2}}a^{-2n-2}bbF(n,n+2,3,{\tfrac {bb}{aa}})\\&A^{\prime \prime \prime }&&={\tfrac {n(n+1)(n+2)}{1.2.3}}a^{-2n-3}b^{3}F(n,n+3,4,{\tfrac {bb}{aa}})\\\end{alignedat}}$

etc.

Si enim $aa+bb-2ab\cos \varphi$ consideratur tamquam productum ex $a-br$ in $a-br^{-1}$ (designante $r$ quantitatem $\cos \varphi +\sin \varphi .{\sqrt {-1}})$, fit $\Omega$ aequalis producto

ex $a^{-2n}$

in $1+n{\tfrac {br}{a}}+{\tfrac {n(n+1)}{1.2}}\cdot {\tfrac {bbrr}{aa}}+{\tfrac {n(n+1)(n+2)}{1.2.3}}\cdot {\tfrac {b^{3}r^{3}}{a^{3}}}+$ etc.

in $1+n{\tfrac {br^{-1}}{a}}+{\tfrac {n(n+1)}{1.2}}\cdot {\tfrac {bbr^{-2}}{aa}}+{\tfrac {n(n+1)(n+2)}{1.2.3}}\cdot {\tfrac {b^{3}r^{-3}}{a^{3}}}+$ etc.

Quod productum quum identicum esse debeat cum

$A+A^{\prime }(r+r^{-1})+A^{\prime \prime }(rr+r^{-2})+A^{\prime \prime \prime }(r^{3}+r^{-3})$

valores supra dati sponte prodeunt.

Porro habemus secundo