Introductio in analysin infinitorum

FRACTARUM EVOLUTIONE. 173 hocque modo progrediendum est donec ultimae fractionis, CAP. XII. cujus denominator est $pp - 2pqz\cos{\Phi} + qqzz$, numerator fuerit determinatus. EXEMPLUM. Sit ista proposita Functio fracta $\frac{z-z^3}{(1+zz)^4(1+z^4)}$ ex cujus denominatoris Factore $(1+zz)^4$ oriantur hae fractiones partiales, $\frac{A+Az}{(1+zz)^4} + \frac{B+Bz}{(1+zz)^3} + \frac{C+Cz}{(1+zz)^2} + \frac{D+Dz}{1+zz}$. Comparatione ergo instituta, erit $p=1$, $q=1$, $\cos{\Phi}=0$; ideoque $\Phi=\frac{1}{2}\pi$, porroque $M=z-z^3$ & $Z=1+z^4$. Hinc erit $M=0; M=2; N=2; N=0, \& \sin{\Phi}=1$. Hinc itaque invenitur $\overline{A}=-\frac{4}{4}\cdot 0 = 0, \& \overline{\overline{A}}=1$. ergo $A+Az=z$; hincque $P = \frac{z- [TRUNCATED]