Page:Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/129

(179)$\left\{{\begin{aligned}&\int _{0}^{\varpi }{\frac {1-r\operatorname {Cos} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.{\frac {\operatorname {l} .\operatorname {Cos} .p}{\left({\frac {p}{2}}\right)^{2}+\left(\operatorname {l} .\operatorname {Cos} .{\frac {p}{2}}\right)^{2}}}\operatorname {d} p\\&\qquad \qquad \qquad \qquad ={\frac {\pi }{1-r}}-{\frac {\pi }{2}}\left[{\frac {1}{\operatorname {l} (2)}}-{\frac {1}{\operatorname {l} (2)-\operatorname {l} \left(1+{\frac {1}{r}}\right)}}\right],\\\\&\int _{0}^{\varpi }{\frac {r\operatorname {Sin} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.{\frac {{\frac {1}{2}}p}{\left({\frac {p}{2}}\right)^{2}+\left(\operatorname {l} .\operatorname {Cos} .{\frac {p}{2}}\right)^{2}}}.\operatorname {d} p\\&\qquad \qquad \qquad \qquad =-{\frac {\pi }{2}}\left[{\frac {1}{\left({\frac {1+r}{2r}}\right)}}+{\frac {1}{\operatorname {l} (2)}}\right]\,;\end{aligned}}\right.$

et ainsi du reste.

Si l’on développe, suivant les puissances ascendantes de $r$, les deux membres de chacune des équations (166), (167), … (172), on obtiendra de nouvelles formules, que l’on pourrait déduire directement de l’équation (122), étendue au cas ou l’équation ${\frac {1}{\varphi (u)}}=0$ a des racines égales. On trouvera de cette manière, $n$ désignant un nombre entier quelconque,

(180)$\left\{{\begin{aligned}&\int _{0}^{\varpi }\operatorname {Cos} .np\left(\operatorname {Tang} .{\frac {p}{2}}\right)^{a}\operatorname {d} p\\\\&=(-1)^{n}.{\frac {\varpi }{2\operatorname {Cos} .{\frac {a\varpi }{2}}}}.{\frac {a(a-1)\ldots (a-n+1)}{1.2.3\ldots n}}\times \\&\left[1+{\frac {n}{1}}.{\frac {a}{a-n+1}}+{\frac {n}{1}}.{\frac {n-1}{2}}.{\frac {a(a+1)}{(a-n+1)(a-n+2)}}+\ldots \right],\\\\&\int _{0}^{\varpi }\operatorname {Sin} .np\left(\operatorname {Tang} .{\frac {p}{2}}\right)^{a}\operatorname {d} p\\\\&=(-1)^{n+1}.{\frac {\varpi }{2\operatorname {Sin} .{\frac {a\varpi }{2}}}}.{\frac {a(a-1)\ldots (a-n+1)}{1.2.3\ldots n}}\times \\&\left[1+{\frac {n}{1}}.{\frac {a}{a-n+1}}+{\frac {n}{1}}.{\frac {n-1}{2}}.{\frac {a(a+1)}{(a-n+1)(a-n+2)}}+\ldots \right].\end{aligned}}\right.$

(181)$\left\{{\begin{aligned}&\int _{0}^{\varpi }\operatorname {Cos} .np.\operatorname {Cos} .{\frac {ap}{2}}\left(\operatorname {Cos} .{\frac {p}{2}}\right)^{-a}\operatorname {d} p\\&\qquad \qquad =(-1)^{n}.2^{a-1}.\varpi .{\frac {a(a+1)\ldots (a+n-1)}{1.2.3\ldots n}},\\\\&\int _{0}^{\varpi }\operatorname {Sin} .np.\operatorname {Sin} .{\frac {ap}{2}}\left(\operatorname {Cos} .{\frac {p}{2}}\right)^{-a}\operatorname {d} p\\&\qquad \qquad =(-1)^{n+1}.2^{a-1}.\varpi .{\frac {a(a+1)\ldots (a+n-1)}{1.2.3\ldots n}}.\end{aligned}}\right.$