Page:Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/123

 Les corrections sont expliquées en page de discussion

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

et ainsi du reste.

Si, au contraire, on prend pour $\varphi (u)$ une fonction rationnelle qui vérifie la condition

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

on trouvera

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

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

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

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

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

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

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