Page:Annales de mathématiques pures et appliquées, 1823-1824, Tome 14.djvu/101

 Les corrections sont expliquées en page de discussion

$S={\frac {1}{2}}(U+V)\,;$

de sorte que tout se réduira à sommer les séries $U$ et $V.$

Pour y parvenir, nous poserons

$\operatorname {Cos} .y={\frac {1}{2}}(p+q),\qquad pq=1\,;$

alors, en faisant

${\begin{aligned}&P={\frac {up}{1}}-{\frac {u^{2}p^{2}}{2}}+{\frac {u^{3}p^{3}}{3}}-{\frac {u^{4}p^{4}}{4}}+\ldots =\operatorname {Log} .(1+up),\\\\&Q={\frac {uq}{1}}-{\frac {u^{2}q^{2}}{2}}+{\frac {u^{3}q^{3}}{3}}-{\frac {u^{4}q^{4}}{4}}+\ldots =\operatorname {Log} .(1+uq),\\\\&P'={\frac {vp}{1}}-{\frac {v^{2}p^{2}}{2}}+{\frac {v^{3}p^{3}}{3}}-{\frac {v^{4}p^{4}}{4}}+\ldots =\operatorname {Log} .(1+vp),\\\\&Q'={\frac {vq}{1}}-{\frac {v^{2}q^{2}}{2}}+{\frac {v^{3}q^{3}}{3}}-{\frac {v^{4}q^{4}}{4}}+\ldots =\operatorname {Log} .(1+vq).\end{aligned}}$

nous aurons

${\begin{aligned}&U={\frac {1}{2}}(P+Q)={\frac {1}{2}}\left\{\operatorname {Log} .(1+up)+\operatorname {Log} .(1+uq)\right\},\\&V={\frac {1}{2}}(P'+Q')={\frac {1}{2}}\left\{\operatorname {Log} .(1+vp)+\operatorname {Log} .(1+vq)\right\}\,;\end{aligned}}$

c’est-à-dire,

${\begin{aligned}&U={\frac {1}{2}}\operatorname {Log} .(1+up)(1+uq)={\frac {1}{2}}\operatorname {Log} .\left[1+(p+q)u+u^{2}\right],\\\\&V={\frac {1}{2}}\operatorname {Log} .(1+vp)(1+vq)={\frac {1}{2}}\operatorname {Log} .\left[1+(p+q)v+v^{2}\right],\end{aligned}}$

ou encore

$2U=\operatorname {Log} .\left(1+2u\operatorname {Cos} .y+u^{2}\right),\qquad 2V=\operatorname {Log} .\left(1+2v\operatorname {Cos} .y+v^{2}\right)\,;$

donc