Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 7.djvu/678

équation par $x^{-{\frac {m}{n}}}dx\,;$ on a généralement

${\begin{aligned}\int &x^{-n-{\frac {m}{n}}}dx\cos(n-2r\pm z)x\\=&-{\frac {\cos(n-2r\pm z)x}{\left(n+{\cfrac {m}{n}}-1\right)x^{n+{\frac {m}{n}}-1}}}+{\frac {(n-2r\pm z)\sin(n-2r\pm z)x}{\left(n+{\cfrac {m}{n}}-1\right)\left(n+{\cfrac {m}{n}}-2\right)x^{n+{\frac {m}{n}}-2}}}\\\\&+{\frac {(n-2r\pm z)^{2}\cos(n-2r\pm z)x}{\left(n+{\cfrac {m}{n}}-1\right)\left(n+{\cfrac {m}{n}}-2\right)\left(n+{\cfrac {m}{n}}-3\right)x^{n+{\frac {m}{n}}-3}}}\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&+{\frac {(-1)^{i}(n-2r\pm z)^{n}}{\left(n+{\cfrac {m}{n}}-1\right)\ldots {\frac {m}{n}}}}\int dxx^{-{\frac {m}{n}}}\cos(n-2r\pm z)x.\end{aligned}}$

On a donc

$\int x^{-{\frac {m}{n}}}dx\cos zx\left({\frac {\sin x}{x}}\right)^{n}={\frac {(-1)^{i}}{2^{2i}x^{n+{\frac {m}{n}}}}}$

$\times \left\{{\begin{aligned}&-{\frac {x}{n+{\cfrac {m}{n}}-1}}\left\{{\begin{aligned}&\cos(n\pm z)x-n\cos(n-2\pm z)x\\+&{\frac {n(n-1)}{1.2}}\cos(n-4\pm z)x\\-&\ldots \ldots \ldots \ldots \ldots \ldots \\\pm &{\frac {1}{2}}{\frac {n(n-1)\ldots (n-i+1)}{1.2.3\ldots i}}\cos(\pm zx)\\\end{aligned}}\right\}\\\\&+{\frac {x^{2}}{\left(n+{\cfrac {m}{n}}-1\right)\left(n+{\cfrac {m}{n}}-2\right)}}\left\{{\begin{aligned}&(n\pm z)\sin(n\pm z)x\\-&n(n-2\pm z)\sin(n-2\pm z)x\\+&\ldots \ldots \ldots \ldots \ldots \ldots \end{aligned}}\right\}\\\\&+{\frac {x^{3}}{\left(n+{\cfrac {m}{n}}-1\right)\left(n+{\cfrac {m}{n}}-2\right)\left(n+{\cfrac {m}{n}}-3\right)}}\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \times \left\{{\begin{aligned}&(n\pm z)^{2}\cos(n\pm z)x\\-&\ldots \ldots \ldots \ldots \ldots \ldots \end{aligned}}\right\}\\&-\ldots \ldots \ldots \ldots \ldots \ldots \end{aligned}}\right\}$

$+{\frac {1}{2^{2i}\left(n+{\cfrac {m}{n}}-1\right)\ldots {\cfrac {m}{n}}}}\int dxx^{-{\frac {m}{n}}}$

$\times \left\{{\begin{aligned}&(n\pm z)^{n}\cos(n\pm z)x\\-&n(n-2\pm z)^{n}\cos(n-2\pm z)x\\+&{\frac {n(n-1)}{1.2}}(n-4\pm z)^{n}\cos(n-4\pm z)x\\+&\ldots \ldots \ldots \ldots \ldots \ldots \\+&{\frac {1}{2}}{\frac {n(n-1)\ldots (n-i-1)}{1.2.3\ldots i}}z^{n}\cos(\pm zx)\end{aligned}}\right\}$

$+\mathrm {const} .$