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Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/122
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{\displaystyle =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)
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{\displaystyle \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}
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{\displaystyle =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)
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{\displaystyle \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}
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{\displaystyle =\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)
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{\displaystyle \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}
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{\displaystyle =\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)
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{\displaystyle \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}
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{\displaystyle =\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)
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{\displaystyle \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}
