Page:Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/130

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La formule (157) et les formules (181), quand on y remplace $a$ par $-a,$ peuvent s’écrire comme il suit :

(182)

\qquad \int _{-\varpi }^{+\varpi }e^{-np{\sqrt {-1}}+be^{p{\sqrt {-1}}}}\operatorname {d} p={\frac {2\varpi }{\Gamma (n+1)}}.b^{n}.

(183)

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

Si, dans les équations (111) et (112), on pose $r=0,$ ou $r=1,$ et $x=\operatorname {Tang} .{\frac {1}{2}}p$ ; on en tirera successivement

(184)

\left\{{\begin{aligned}&\int _{0}^{\varpi }\left(\operatorname {Tang} .{\frac {p}{2}}\right)^{-a}\operatorname {Cos} .\left(b\operatorname {Tang} .{\frac {p}{2}}\right){\frac {\operatorname {d} p}{\operatorname {Cos} .^{2}{\frac {p}{2}}}}={\frac {\varpi }{\Gamma (a).\operatorname {Cos} .{\frac {a\pi }{2}}}}.b^{a-1},\\\\&\int _{0}^{\varpi }\left(\operatorname {Tang} .{\frac {p}{2}}\right)^{-a}\operatorname {Sin} .\left(b\operatorname {Tang} .{\frac {p}{2}}\right){\frac {\operatorname {d} p}{\operatorname {Cos} .^{2}{\frac {p}{2}}}}={\frac {\varpi }{\Gamma (a).\operatorname {Sin} .{\frac {a\pi }{2}}}}.b^{a-1}.\end{aligned}}\right.

(185)

\left\{{\begin{aligned}&\int _{0}^{\varpi }\operatorname {Cos} .{\frac {ap}{2}}\left(\operatorname {Cos} .{\frac {p}{2}}\right)^{a-2}\operatorname {Cos} .\left(b\operatorname {Tang} .{\frac {p}{2}}\right)\operatorname {d} p={\frac {\pi }{\Gamma (a)}}.b^{a-1}.e^{a-1},\\\\&\int _{0}^{\varpi }\operatorname {Sin} .{\frac {ap}{2}}\left(\operatorname {Cos} .{\frac {p}{2}}\right)^{a-2}\operatorname {Sin} .\left(b\operatorname {Tang} .{\frac {p}{2}}\right)\operatorname {d} p={\frac {\pi }{\Gamma (a)}}.b^{a-1}.e^{-b}.\end{aligned}}\right.

(186)

\left\{{\begin{aligned}&\int _{0}^{\varpi }\left(\operatorname {Sin} .{\frac {p}{2}}\right)^{-a}\left(\operatorname {Cos} .{\frac {p}{2}}\right)^{a+b-2}.\operatorname {Cos} .{\frac {bp}{2}}\operatorname {d} p={\frac {\varpi }{\operatorname {Cos} .{\frac {a\varpi }{2}}}}.{\frac {\Gamma (a+b-1)}{\Gamma (a).\Gamma (b)}},\\\\&\int _{0}^{\varpi }\left(\operatorname {Sin} .{\frac {p}{2}}\right)^{-a}\left(\operatorname {Sin} .{\frac {p}{2}}\right)^{a+b-2}.\operatorname {Sin} .{\frac {bp}{2}}\operatorname {d} p={\frac {\pi }{\operatorname {Sin} .{\frac {a\pi }{2}}}}.{\frac {\Gamma (a+b-1)}{\Gamma (a).\Gamma (b)}}.\end{aligned}}\right.