Page:Annales de mathématiques pures et appliquées, 1824-1825, Tome 15.djvu/187

${\frac {1}{2\varpi }}\int _{0}^{\varpi }{\frac {\operatorname {Sin} .rx}{\operatorname {Sin} .x}}\left(e^{me^{x{\sqrt {-1}}}}+e^{me^{-x{\sqrt {-1}}}}\right)\operatorname {d} x\,;$

ou bien

${\frac {1}{2\varpi }}\int _{0}^{\varpi }{\frac {\operatorname {Sin} .rx}{\operatorname {Sin} .x}}\left\{e^{m\left(\operatorname {Cos} .x+{\sqrt {-1}}\operatorname {Sin} .x\right)}+e^{m\left(\operatorname {Cos} .x-{\sqrt {-1}}\operatorname {Sin} .x\right)}\right\}\operatorname {d} x,$

ou encore

${\frac {1}{2\varpi }}\int _{0}^{\varpi }{\frac {\operatorname {Sin} .rx}{\operatorname {Sin} .x}}.e^{m\operatorname {Cos} .x}.\left(e^{m\operatorname {Sin} .x{\sqrt {-1}}}+e^{-m\operatorname {Sin} .x{\sqrt {-1}}}\right)\operatorname {d} x,$

ou enfin

${\frac {1}{2\varpi }}\int _{0}^{\varpi }{\frac {\operatorname {Sin} .rx}{\operatorname {Sin} .x}}.e^{m\operatorname {Cos} .x}.\operatorname {Cos} .(m\operatorname {Sin} .x)\operatorname {d} x.$

On aura, en outre, en faisant toujours $\alpha =0,$ après la différentiation,

$\operatorname {F} (\alpha )=1,\quad \operatorname {F} '(\alpha )=m,\quad \operatorname {F} ''(\alpha )=m^{2},\ldots \operatorname {F} ^{(r-1)}(\alpha )=m^{r-1}\,;$

en conséquence l’équation (A) deviendra

${\frac {1}{2\varpi }}\int _{0}^{\varpi }{\frac {\operatorname {Sin} .rx}{\operatorname {Sin} .x}}.e^{m\operatorname {Cos} .x}.\operatorname {Cos} .(m\operatorname {Sin} .x)\operatorname {d} x$

$={\frac {m^{r-1}}{1.2\ldots (r-1)}}+{\frac {m^{r-3}}{1.2\ldots (r-3)}}+{\frac {m^{r-5}}{1.2\ldots (r-5)}}+\ldots$

c’est-à-dire, si $r$ est un nombre pair,

${\frac {1}{\varpi }}\int _{0}^{\varpi }{\frac {\operatorname {Sin} .rx}{\operatorname {Sin} .x}}.e^{m\operatorname {Cos} .x}.\operatorname {Cos} .(m\operatorname {Sin} .x)\operatorname {d} x$

$={\frac {m}{1}}+{\frac {m^{3}}{1.2.3}}+{\frac {m^{5}}{1.2.3.4.5}}+\ldots {\frac {m^{r-1}}{1.2\ldots (r-1)}}\,;$

et si $r$ est un nombre impair,

${\frac {1}{\varpi }}\int _{0}^{\varpi }{\frac {\operatorname {Sin} .rx}{\operatorname {Sin} .x}}.e^{m\operatorname {Cos} .x}.\operatorname {Cos} .(m\operatorname {Sin} .x)\operatorname {d} x$

$=1+{\frac {m^{2}}{1.2}}+{\frac {m^{4}}{1.2.3.4}}+\ldots {\frac {m^{r-1}}{1.2\ldots (r-1)}},$