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Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/120
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{\displaystyle =2\varpi \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\}.}
(136)
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{\displaystyle \int _{-\varpi }^{+\varpi }{\frac {\operatorname {Cos} .{\frac {ap}{2}}-{\sqrt {-1}}\operatorname {Sin} .{\frac {ap}{2}}}{\left(\operatorname {Cos} .{\frac {p}{2}}\right)^{a}}}\varphi \left(e^{p{\sqrt {-1}}}\right)\operatorname {d} p}
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{\displaystyle =2^{a+1}.\varpi \left\{\varphi (0)+{\frac {H+K{\sqrt {-1}}}{h+k{\sqrt {-1}}}}\left(1+h+k{\sqrt {-1}}\right)^{-a}+\ldots \right\}.}
(137)
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{\displaystyle \int _{-\varpi }^{+\varpi }\left(1-{\sqrt {-1}}\operatorname {Cot} .{\frac {p}{2}}\right)^{a}\varphi \left(e^{p{\sqrt {-1}}}\right)\operatorname {d} p}
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{\displaystyle =2^{a+1}.\varpi \left\{\varphi (0)+{\frac {H+K{\sqrt {-1}}}{h+k{\sqrt {-1}}}}\left(1-h-k{\sqrt {-1}}\right)^{-a}+\ldots \right\}\,;}
(138)
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{\displaystyle \int _{-\varpi }^{+\varpi }\operatorname {l} \left(-{\sqrt {-1}}.\operatorname {Tang} .{\frac {p}{2}}\right)\varphi \left(e^{p{\sqrt {-1}}}\right)\operatorname {d} p}
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{\displaystyle =2\varpi \left\{{\frac {H+K{\sqrt {-1}}}{h+k{\sqrt {-1}}}}.\left({\frac {1-h-k{\sqrt {-1}}}{1+h+k{\sqrt {-1}}}}\right)+\ldots \right\}.}
(139)
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{\displaystyle \int _{-\varpi }^{+\varpi }\left[\operatorname {l} \operatorname {Cos} .{\frac {p}{2}}+{\frac {1}{2}}p{\sqrt {-1}}\right]\varphi \left(e^{p{\sqrt {-1}}}\right)\operatorname {d} p}
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{\displaystyle =2\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\}.}
(140)
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{\displaystyle \int _{-\varpi }^{+\varpi }\operatorname {l} \left(1+{\sqrt {-1}}.\operatorname {Cot} .{\frac {p}{2}}\right).\varphi \left(e^{p{\sqrt {-1}}}\right)\operatorname {d} p}
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{\displaystyle =-2\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\}.}
![{\displaystyle \int _{-\varpi }^{+\varpi }\left[\operatorname {l} \operatorname {Cos} .{\frac {p}{2}}+{\frac {1}{2}}p{\sqrt {-1}}\right]\varphi \left(e^{p{\sqrt {-1}}}\right)\operatorname {d} p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80aecbba1943f0e3ee0b57778660370e42d8e755)
