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Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/111
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(97)
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{\displaystyle \int _{-\infty }^{+\infty }\left(-x{\sqrt {-1}}\right)^{a-1}e^{bx{\sqrt {-1}}}.\operatorname {l} \left(1+{\frac {s}{x}}{\sqrt {-1}}\right){\frac {\operatorname {d} x}{r-x{\sqrt {-1}}}}=0.}
(98)
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{\displaystyle \int _{-\infty }^{+\infty }\left(-x{\sqrt {-1}}\right)^{a-1}e^{bx{\sqrt {-1}}}.\operatorname {l} \left(1+{\frac {s}{x}}{\sqrt {-1}}\right){\frac {\operatorname {d} x}{r+x{\sqrt {-1}}}}=2\varpi .r^{a-1}e^{-br}.\operatorname {l} \left(1+{\frac {s}{r}}\right).}
(99)
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{\displaystyle \left\{{\begin{aligned}&\int _{-\infty }^{+\infty }{\frac {e^{ax{\sqrt {-1}}}-e^{bx{\sqrt {-1}}}}{x{\sqrt {-1}}}}.{\frac {\operatorname {d} x}{\operatorname {l} \left(r-x{\sqrt {-1}}\right)}}=\\&\qquad \qquad \qquad \qquad -{\frac {2\varpi }{1-r}}\left[e^{-b(1-r)}-e^{-a(1-r)}\right],\mathrm {\ pour\ } r<1\\\\&\int _{-\infty }^{+\infty }{\frac {e^{ax{\sqrt {-1}}}-e^{bx{\sqrt {-1}}}}{x{\sqrt {-1}}}}.{\frac {\operatorname {d} x}{\operatorname {l} \left(1-x{\sqrt {-1}}\right)}}=-\varpi (a-b)\,;\\\\&\int _{-\infty }^{+\infty }{\frac {e^{ax{\sqrt {-1}}}-e^{bx{\sqrt {-1}}}}{x{\sqrt {-1}}}}.{\frac {\operatorname {d} x}{\operatorname {l} \left(r-x{\sqrt {-1}}\right)}}=0,\mathrm {\ pour\ } r>1.\end{aligned}}\right.}
(100)
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{\displaystyle \int _{-\infty }^{+\infty }\left(-x{\sqrt {-1}}\right)^{a-1}e^{bx{\sqrt {-1}}}.{\frac {\operatorname {l} \left(1+{\frac {s}{x}}{\sqrt {-1}}\right)}{\operatorname {l} \left(1-{\frac {s}{x}}{\sqrt {-1}}\right)}}\operatorname {d} x}
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{\displaystyle =2\varpi (1-r)^{a-1}e^{-b(1-r)}\operatorname {l} \left({\frac {1-r}{1-r+s}}\right)\,;\mathrm {\ pour\ } r<1}
(101)
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{\displaystyle \int _{0}^{\infty }{\frac {{\frac {\pi }{2}}(\operatorname {Cos} .ax-\operatorname {Cos} .bx)+(\operatorname {Sin} .ax-\operatorname {Sin} .bx)\operatorname {l} (x)}{\left({\frac {\varpi }{2}}\right)^{2}+\left[\operatorname {l} (x)\right]^{2}}}.{\frac {\operatorname {d} x}{x}}=\varpi \left(e^{-a}-e^{-b}\right).}
(102)
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{\displaystyle \int _{-\infty }^{+\infty }{\frac {e^{bx}{\sqrt {-1}}}{\left(r-x{\sqrt {-1}}\right)^{a}}}\operatorname {d} x=0,\quad \int _{-\infty }^{+\infty }{\frac {\operatorname {d} x}{\left(r-x{\sqrt {-1}}\right)^{a}\left(s-x{\sqrt {-1}}\right)^{b}}}=0.}
(103)
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{\displaystyle \int _{0}^{\infty }e^{a\operatorname {Cos} .bx}(a\operatorname {Sin} .bx){\frac {\operatorname {d} x}{x}}={\frac {\varpi }{2}}\left(e^{a}-1\right).}
![{\displaystyle \left\{{\begin{aligned}&\int _{-\infty }^{+\infty }{\frac {e^{ax{\sqrt {-1}}}-e^{bx{\sqrt {-1}}}}{x{\sqrt {-1}}}}.{\frac {\operatorname {d} x}{\operatorname {l} \left(r-x{\sqrt {-1}}\right)}}=\\&\qquad \qquad \qquad \qquad -{\frac {2\varpi }{1-r}}\left[e^{-b(1-r)}-e^{-a(1-r)}\right],\mathrm {\ pour\ } r<1\\\\&\int _{-\infty }^{+\infty }{\frac {e^{ax{\sqrt {-1}}}-e^{bx{\sqrt {-1}}}}{x{\sqrt {-1}}}}.{\frac {\operatorname {d} x}{\operatorname {l} \left(1-x{\sqrt {-1}}\right)}}=-\varpi (a-b)\,;\\\\&\int _{-\infty }^{+\infty }{\frac {e^{ax{\sqrt {-1}}}-e^{bx{\sqrt {-1}}}}{x{\sqrt {-1}}}}.{\frac {\operatorname {d} x}{\operatorname {l} \left(r-x{\sqrt {-1}}\right)}}=0,\mathrm {\ pour\ } r>1.\end{aligned}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b22cc58d2b4730e6a3835da047faf5598045320c)
![{\displaystyle \int _{0}^{\infty }{\frac {{\frac {\pi }{2}}(\operatorname {Cos} .ax-\operatorname {Cos} .bx)+(\operatorname {Sin} .ax-\operatorname {Sin} .bx)\operatorname {l} (x)}{\left({\frac {\varpi }{2}}\right)^{2}+\left[\operatorname {l} (x)\right]^{2}}}.{\frac {\operatorname {d} x}{x}}=\varpi \left(e^{-a}-e^{-b}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7186b81effa54740b91446a74e92d386c62527ae)
