Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/392

380

SOMMATION

${\frac {1}{2{\sqrt {-1}}}}\operatorname {Log} .{\frac {(1+a\operatorname {Cos} .t)+{\sqrt {-1}}a\operatorname {Sin} .t}{(1+a\operatorname {Cos} .t)-{\sqrt {-1}}a\operatorname {Sin} .t}}.$

En vertu de la formule connue

$\operatorname {Log} .{\frac {p}{q}}=2\left\{{\frac {p-q}{p+q}}+{\frac {1}{3}}\left({\frac {p-q}{p+q}}\right)^{3}+{\frac {1}{5}}\left({\frac {p-q}{p+q}}\right)^{5}+\ldots \right\},$

cette somme devient

${\frac {a\operatorname {Sin} .t}{1+a\operatorname {Cos} .t}}-{\frac {1}{3}}\left({\frac {a\operatorname {Sin} .t}{1+a\operatorname {Cos} .t}}\right)^{3}+{\frac {1}{5}}\left({\frac {a\operatorname {Sin} .t}{1+a\operatorname {Cos} .t}}\right)^{5}-\ldots$

c’est-à-dire,

$\operatorname {Arc} .\left(\operatorname {Tang} .={\frac {a\operatorname {Sin} .t}{1+a\operatorname {Cos} .t}}\right).$

17. D’après cela, on aura ici

$\operatorname {F} [t]=2\operatorname {Log} .{\sqrt {1+2a\operatorname {Cos} .t+a^{2}}},\ \operatorname {F} '[t]=2\operatorname {Arc} .\left(\operatorname {Tang} .={\frac {a\operatorname {Sin} .t}{1+a\operatorname {Cos} .t}}\right)\,;$

substituant donc ces valeurs dans les formules ci-dessus (9, 10), nous aurons, par le Théorème I, quel que soit $n$,

${\frac {a\operatorname {Cos} .t\operatorname {Cos} .u\operatorname {Cos} .v\ldots }{1}}-{\frac {a^{2}\operatorname {Cos} .2t\operatorname {Cos} .2u\operatorname {Cos} .2v\ldots }{2}}+{\frac {a^{3}\operatorname {Cos} .3t\operatorname {Cos} .3u\operatorname {Cos} .3v\ldots }{3}}-\ldots$

$=+{\frac {1}{2^{n-1}}}\left\{\operatorname {Log} .{\sqrt {1+2a\operatorname {Cos} .s+a^{2}}}+\Sigma \operatorname {Log} .{\sqrt {1+2a\operatorname {Cos} .(s-2t)+a^{2}}}\right.$

$\left.+\Sigma \operatorname {Log} .{\sqrt {1+2a\operatorname {Cos} .(s-2t-2u)+a^{2}}}+\ldots \right\}$

par le Théorème II, pour $n$ pair,

${\frac {a\operatorname {Sin} .t\operatorname {Sin} .u\operatorname {Sin} .v\ldots }{1}}-{\frac {a^{2}\operatorname {Sin} .2t\operatorname {Sin} .2u\operatorname {Sin} .2v\ldots }{2}}+{\frac {a^{3}\operatorname {Sin} .3t\operatorname {Sin} .3u\operatorname {Sin} .3v\ldots }{3}}-\ldots$