Page:Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/122

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

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

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

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

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

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

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

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

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

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