Page:Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/121

(141)$\int _{-\varpi }^{+\varpi }{\frac {\operatorname {l} \operatorname {Cos} .{\frac {p}{2}}-{\frac {p}{2}}{\sqrt {-1}}}{\left(\operatorname {l} \operatorname {Cos} .{\frac {p}{2}}\right)^{2}+\left({\frac {p}{2}}\right)^{2}}}.\varphi \left(e^{p{\sqrt {-1}}}\right)\operatorname {d} p$

$=-2\varpi \left\{{\frac {\varphi (0)}{\operatorname {l} (2)}}-\varphi (1)+{\frac {H+K{\sqrt {-1}}}{h+k{\sqrt {-1}}}}.{\frac {1}{\operatorname {l} (2)-\operatorname {l} \left(1+h+k{\sqrt {-1}}\right)}}+\ldots \right\}.$

(142)$\int _{-\varpi }^{+\varpi }\left(-{\sqrt {-1}}.\operatorname {Tang} .{\frac {p}{2}}\right)^{a}{\frac {\operatorname {l} \operatorname {Cos} .{\frac {p}{2}}-{\frac {p}{2}}{\sqrt {-1}}}{\left(\operatorname {l} \operatorname {Cos} .{\frac {p}{2}}\right)^{2}+\left({\frac {p}{2}}\right)^{2}}}.\varphi \left(e^{p{\sqrt {-1}}}\right)\operatorname {d} p$

$=-2\varpi \left\{{\frac {\varphi (0)}{\operatorname {l} (2)}}+{\frac {H+K{\sqrt {-1}}}{h+k{\sqrt {-1}}}}\left({\frac {1-h-k{\sqrt {-1}}}{1+h+k{\sqrt {-1}}}}\right)^{a}{\frac {1}{\operatorname {l} (2)-\operatorname {l} \left(1+h+k{\sqrt {-1}}\right)}}+\ldots \right\}\,;$

et ainsi du reste.

Par suite, si l’on prend pour $\varphi (u)$ une fonction réelle de $u$ qui vérifie la condition

(143)$\qquad \varphi (u)=\varphi \left({\frac {1}{u}}\right),$

on trouvera

(144)$\qquad \int _{0}^{\varpi }{\frac {\varphi \left(e^{p{\sqrt {-1}}}\right)+\varphi \left(e^{-p{\sqrt {-1}}}\right)}{2}}.\left(\operatorname {Tang} .{\frac {p}{2}}\right)^{a}\operatorname {d} p$

$={\frac {\pi }{\operatorname {Cos} .{\frac {a\pi }{2}}}}\left\{\varphi (0)+{\frac {H+K{\sqrt {-1}}}{h+k{\sqrt {-1}}}}\left({\frac {1-h-k{\sqrt {-1}}}{1+h+k{\sqrt {-1}}}}\right)^{a}+\ldots \right\}.$

(145)$\qquad \int _{0}^{\varpi }{\frac {\varphi \left(e^{p{\sqrt {-1}}}\right)+\varphi \left(e^{-p{\sqrt {-1}}}\right)}{2}}.{\frac {\operatorname {Cos} .{\frac {ap}{2}}}{\left(\operatorname {Cos} .{\frac {p}{2}}\right)^{a}}}\operatorname {d} p$