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Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/99
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(32)
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{\displaystyle \int _{-\infty }^{+\infty }\operatorname {l} \left(1-rx{\sqrt {-1}}\right)\varphi (x)\operatorname {d} x=}
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{\displaystyle -2\varpi \left[\left(K-H{\sqrt {-1}}\right)\operatorname {l} \left(1+kr-hr{\sqrt {-1}}\right)+\ldots \right].}
(33)
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{\displaystyle \int _{-\infty }^{+\infty }\operatorname {l} \left[r\operatorname {Sin} .\theta +\left(r\operatorname {Cos} .\theta -x\right){\sqrt {-1}}\right]\varphi (x)\operatorname {d} x}
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{\displaystyle =-2\varpi \left\{\left(K-H{\sqrt {-1}}\right)\operatorname {l} \left[k+r\operatorname {Sin} .\theta -(k-r\operatorname {Cos} .\theta {\sqrt {-1}}){\sqrt {-1}}\right]+\ldots \right\}.}
(34)
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{\displaystyle \int _{-\infty }^{+\infty }\left(-x{\sqrt {-1}}\right)^{a-1}e^{bx{\sqrt {-1}}}\varphi (x)\operatorname {d} x}
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{\displaystyle =-2\varpi \left\{\left(K-H{\sqrt {-1}}\right)\left(k-h{\sqrt {-1}}\right)^{a-1}e^{-bx}(\operatorname {Cos} .bh+{\sqrt {-1}}\operatorname {Sin} .bh)+\ldots \right\}.}
(35)
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{\displaystyle \int _{-\infty }^{+\infty }{\frac {\left(-x{\sqrt {-1}}\right)^{a}}{\operatorname {l} \left(1-rx{\sqrt {-1}}\right)}}\varphi (x)\operatorname {d} x}
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{\displaystyle =-2\varpi \left\{\left(K-H{\sqrt {-1}}\right)\left(k+h{\sqrt {-1}}\right)^{a}{\frac {1}{\operatorname {l} \left(1+kr-hr{\sqrt {-1}}\right)}}+\ldots \right\}.}
(36)
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{\displaystyle \int _{-\infty }^{+\infty }\left(-x{\sqrt {-1}}\right)^{a-1}.\operatorname {l} \left(1+{\frac {s}{x}}{\sqrt {-1}}\right)\varphi (x)\operatorname {d} x}
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{\displaystyle =-2\varpi \left\{\left(K-H{\sqrt {-1}}\right)\left(k-h{\sqrt {-1}}\right)^{a-1}\operatorname {l} \left(1+{\frac {s}{\rho }}e^{\omega {\sqrt {-1}}}\right)+\ldots \right\}.}
(37)
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{\displaystyle \int _{-\infty }^{+\infty }{\frac {\left(-x{\sqrt {-1}}\right)^{a}}{\operatorname {l} \left(1-rx{\sqrt {-1}}\right)}}.e^{bx{\sqrt {-1}}}.\operatorname {l} \left(1+{\frac {s}{x}}{\sqrt {-1}}\right)\varphi (x)\operatorname {d} x}
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{\displaystyle =-2\varpi \left\{\left(K-H{\sqrt {-1}}\right)\left(k-h{\sqrt {-1}}\right)^{a}\operatorname {l} \left(1+{\frac {s}{\rho }}e^{\omega {\sqrt {-1}}}\right){\frac {1}{\operatorname {l} \left(1+kr-hr{\sqrt {-1}}\right)}}+\ldots \right\}.}
![{\displaystyle -2\varpi \left[\left(K-H{\sqrt {-1}}\right)\operatorname {l} \left(1+kr-hr{\sqrt {-1}}\right)+\ldots \right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05819146d6b63582b02f8382a89fbab10fc11d42)
![{\displaystyle \int _{-\infty }^{+\infty }\operatorname {l} \left[r\operatorname {Sin} .\theta +\left(r\operatorname {Cos} .\theta -x\right){\sqrt {-1}}\right]\varphi (x)\operatorname {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de1d9ea8951950ab6bc99618ad7f8ec391ec7e22)
![{\displaystyle =-2\varpi \left\{\left(K-H{\sqrt {-1}}\right)\operatorname {l} \left[k+r\operatorname {Sin} .\theta -(k-r\operatorname {Cos} .\theta {\sqrt {-1}}){\sqrt {-1}}\right]+\ldots \right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61946c999fc19d38fe76d029dfa980273f060ba4)
