dans la seconde, il vient
![{\displaystyle {\begin{aligned}&{\frac {8}{\pi }}\int _{0}^{\infty }\left[\cos(\mu -2n)gx{\cos }\gamma x{\sin }\varepsilon x-{\frac {{\cos }\alpha (\mu -2n)gx{\cos }\alpha \gamma x{\sin }\alpha \varepsilon x}{\alpha ^{\mu -1}}}+(1-\alpha )\mathrm {E} \right]{\frac {dx}{x^{\mu +1}}}\\&\qquad =\pm {\frac {(1-\alpha )(-1)^{{\frac {1}{2}}\mu }}{1.2.3\!\ldots \!\mu }}[\pm (\gamma +\mu g-2ng+\varepsilon )^{\mu }\pm (\gamma -\mu g+2ng+\varepsilon )^{\mu }\\&\qquad \mp (\gamma +\mu g-2ng-\varepsilon )^{\mu }\mp (\gamma -\mu g+2ng-\varepsilon )^{\mu }],\\[1em]&{\frac {8}{\pi }}\int _{0}^{\infty }\left[\sin(\mu -2n)gx{\cos }\gamma x{\sin }\varepsilon x-{\frac {{\sin }\alpha (\mu -2n)gx{\cos }\alpha \gamma x{\sin }\alpha \varepsilon x}{\alpha ^{\mu -1}}}+(1-\alpha )\mathrm {E'} \right]{\frac {dx}{x^{\mu +1}}}\\&\qquad =\pm {\frac {(1-\alpha )(-1)^{{\frac {1}{2}}(\mu -1)}}{1.2.3\!\ldots \!\mu }}[\pm (\gamma +\mu g-2ng+\varepsilon )^{\mu }\mp (\gamma -\mu g+2ng+\varepsilon )^{\mu }\\&\qquad \mp (\gamma +\mu g-2ng-\varepsilon )^{\mu }\pm (\gamma -\mu g+2ng-\varepsilon )^{\mu }]\;;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e958989cefc64908bf5d869243c5d2001c5e077)
et
désignant aussi des constantes différentes de
et
. En donnant à
les valeurs successives 0, 1, 2, 3, etc. ; faisant, pour abréger,
![{\displaystyle {\begin{aligned}u&=\left[\cos {\mu gx}-\mu \cos {(\mu -2)gx}+{\frac {\mu \,{.}\,\mu {-}1}{1\,{.}\,2}}\cos {(\mu -4)gx}\right.\\&\left.{}-{\frac {\mu \,{.}\,\mu {-}1\,{.}\,\mu {-}2}{1\,{.}\,2\,{.}\,3}}\cos {(\mu -6)gx}+{\text{etc.}}\right]{\frac {\cos {\gamma x}\,\sin {\varepsilon x}}{x^{\mu -1}}}\;;\\v&=\left[\sin {\mu gx}-\mu \sin {(\mu -2)gx}+{\frac {\mu \,{.}\,\mu {-}1}{1\,{.}\,2}}\sin {(\mu -4)gx}\right.\\&\left.{}-{\frac {\mu \,{.}\,\mu {-}1\,{.}\,\mu {-}2}{1\,{.}\,2\,{.}\,3}}\sin {(\mu -6)gx}+{\text{etc.}}\right]{\frac {\cos {\gamma x}\,\sin {\varepsilon x}}{x^{\mu -1}}}\;;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9b0c6ba9c4a34ae016b137857a15bb976305c3e)
et désignant par
et
ce que deviennent
et
, quand on y change
en
, on déduit des équations précédentes
![{\displaystyle {\begin{aligned}{\frac {8}{\pi }}\int _{0}^{\infty }\left[u-u'+{\frac {(1{-}\alpha )\mathrm {F} }{x^{\mu -1}}}\right]\,\!\!{\frac {dx}{x^{2}}}&={\frac {(1{-}\alpha )(-1)^{{\frac {1}{2}}\mu }}{1\,{.}\,2\,{.}\,3\ldots \mu }}(\Gamma {+}\Gamma '{-}\Gamma _{\prime }{-}\Gamma '_{\prime }),\\{\frac {8}{\pi }}\int _{0}^{\infty }\left[v-v'+{\frac {(1{-}\alpha )\mathrm {F'} }{x^{\mu -1}}}\right]\!\!{\frac {dx}{x^{2}}}&={\frac {(1{-}\alpha )(-1)^{{\frac {1}{2}}(\mu -1)}}{1\,{.}\,2\,{.}\,3\ldots \mu }}(\Gamma {-}\Gamma '{-}\Gamma _{\prime }{+}\Gamma '_{\prime })\;;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3db8dba50cab8c786e864b935c0cc6446e3b51f)