![{\displaystyle {\frac {1}{2\varpi {\sqrt {-1}}}}\int _{0}^{\varpi }\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/db70ef2592e0f6a8aa8a17345c8af6e828f55baa)
![{\displaystyle {\frac {\operatorname {F} \left(\alpha +pe^{x{\sqrt {-1}}}\right).\left(\operatorname {Cos} .rx+{\sqrt {-1}}\operatorname {Sin} .rx\right)-\operatorname {F} \left(\alpha +pe^{-x{\sqrt {-1}}}\right).\left(\operatorname {Cos} .rx-{\sqrt {-1}}\operatorname {Sin} .rx\right)}{\operatorname {Sin} .x}}\operatorname {d} x\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d27f68ee44246ee5b8c37ea0a0711ffa2eb649f)
ou encore
![{\displaystyle {\frac {1}{2\varpi }}\int _{0}^{\varpi }{\frac {e^{rx{\sqrt {-1}}}.\operatorname {F} \left(\alpha +pe^{x{\sqrt {-1}}}\right)-e^{-rx{\sqrt {-1}}}.\operatorname {F} \left(\alpha +pe^{-x{\sqrt {-1}}}\right)}{{\sqrt {-1}}.\operatorname {Sin} .x}}\operatorname {d} x\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72fd304119f40570aea74e343ed052caa6e5929f)
de sorte qu’on pourra écrire, si
est un nombre pair,
![{\displaystyle {\frac {1}{\varpi }}\int _{0}^{\varpi }{\frac {e^{rx{\sqrt {-1}}}.\operatorname {F} \left(\alpha +pe^{x{\sqrt {-1}}}\right)-e^{-rx{\sqrt {-1}}}.\operatorname {F} \left(\alpha +pe^{-x{\sqrt {-1}}}\right)}{{\sqrt {-1}}.\operatorname {Sin} .x}}\operatorname {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e26ea921810b76d75b723bbc498ab39f59a2c1ca)
![{\displaystyle =\operatorname {F} (\alpha +p)-\operatorname {F} (\alpha -p)\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d41c46b6288e6bde2016c73ba0b663557d58ed)
(E)
et, si
est un nombre impair,
![{\displaystyle {\frac {1}{\varpi }}\int _{0}^{\varpi }{\frac {e^{rx{\sqrt {-1}}}.\operatorname {F} \left(\alpha +pe^{x{\sqrt {-1}}}\right)-e^{-rx{\sqrt {-1}}}.\operatorname {F} \left(\alpha +pe^{-x{\sqrt {-1}}}\right)}{{\sqrt {-1}}.\operatorname {Sin} .x}}\operatorname {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e26ea921810b76d75b723bbc498ab39f59a2c1ca)
![{\displaystyle =\operatorname {F} (\alpha +p)+\operatorname {F} (\alpha -p)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f92461ccc6d64a85449611d818f3fb41290ec97)
(F)
On remarquera que les seconds membres de ces équations sont indépendans de ![{\displaystyle r.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10110093812676dd04a92ce4c8b75940c366330a)
Si dans la formule (B), on fait tour-à-tour
et
elle donnera
![{\displaystyle {\frac {1}{\varpi }}\int _{0}^{\varpi }{\frac {\operatorname {F} \left(\alpha +pe^{x{\sqrt {-1}}}\right)-\operatorname {F} \left(\alpha +pe^{-x{\sqrt {-1}}}\right)}{{\sqrt {-1}}.\operatorname {Sin} .x}}\operatorname {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28201640a39d60a2e83a51e5869a508f92189864)
![{\displaystyle =\operatorname {F} (\alpha +p)-\operatorname {F} (\alpha -p)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e7ee62e751a2390ae5c0d92563ce9313568ecec)
(G)
![{\displaystyle {\frac {1}{\varpi }}\int _{0}^{\varpi }{\frac {\left\{\operatorname {F} \left(\alpha +pe^{x{\sqrt {-1}}}\right)-\operatorname {F} \left(\alpha +pe^{-x{\sqrt {-1}}}\right)\right\}\operatorname {Cos} .x}{{\sqrt {-1}}.\operatorname {Sin} .x}}\operatorname {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cffea855dfbe262252cadee990e67002fd353ad)
![{\displaystyle =\operatorname {F} (\alpha +p)+\operatorname {F} (\alpha -p)-2\operatorname {F} (\alpha )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e99ef3931a871295cfee228aa4b8ccb5aa6eee2a)
(H)
mais on ne pourrait obtenir des résultats analogues de la formule (A) qu’en y supposant
infini.