en deux équations réelles, lorsqu’on égale séparément à zéro et les parties réelles des deux membres vet les parties multipliées par
En opérant ainsi, et prenant pour
une fonction réelle, on obtiendra une multitude de formules, dont quelques-unes sont déjà connues, et parmi lesquelles je citerai seulement les suivantes :
(46)
![{\displaystyle \int _{0}^{\infty }x^{a-1}\varphi (x)\operatorname {d} x=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ecac92675fba3564191be3f5702d2f8273e7caa)
![{\displaystyle {\frac {2\varpi }{\operatorname {Sin} .a\varpi }}\left\{\rho ^{a-1}\left[+k\operatorname {Cos} .(1-a)\left({\frac {\pi }{2}}+\omega \right)-h\operatorname {Sin} .(1-a)\left({\frac {\pi }{2}}+\omega \right)\right]+\ldots \right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cdfd0d4104e43f4eb1e91d183d23722d2fe895b)
(47)
![{\displaystyle \int _{-\infty }^{+\infty }\operatorname {Cos} .bx\varphi (x)\operatorname {d} x=-2\varpi \left\{\left(K\operatorname {Cos} .bh+H\operatorname {Sin} .bh\right)e^{-bk}+\ldots \right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b578aa7193a05464d11852762c4d85a5c71e3ae)
(48)
![{\displaystyle \int _{-\infty }^{+\infty }\operatorname {Sin} .bx\varphi (x)\operatorname {d} x=-2\varpi \left\{\left(K\operatorname {Sin} .bh-H\operatorname {Cos} .bh\right)e^{-bk}+\ldots \right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15059e4c8afbf73146baa02ddd85fc276b7e977a)
(49)
![{\displaystyle \int _{-\infty }^{+\infty }\operatorname {l} \left(r^{2}-2rx\operatorname {Cos} .\theta +x^{2}\right)\varphi (x)\operatorname {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d5d2f93e91c68f2eee91486591e9334f306e176)
![{\displaystyle =-2\varpi \left\{K\operatorname {l} \left[r^{2}-2\rho r\operatorname {Sin} .(\omega -\theta )+\rho ^{2}\right]-2H\operatorname {Arc} .\operatorname {Tang} .={\frac {\rho \operatorname {Sin} .\omega -r\operatorname {Cos} .\theta }{\rho \operatorname {Cos} .\omega +r\operatorname {Sin} .\theta }}+\ldots \right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11197ac2400d813a71685b4674944b5185b0f0d1)
(50)
![{\displaystyle \int _{-\infty }^{+\infty }\operatorname {Arc} .\operatorname {Tang} .={\frac {r\operatorname {Cos} .\theta -x}{r\operatorname {Sin} .\theta }}.\varphi (x)\operatorname {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb1eff7a40d69ff463c62e9a4ac2a30b131160c)
![{\displaystyle =\varpi \left\{H\operatorname {l} \left[r^{2}-2\rho r\operatorname {Sin} .(\omega -\theta )+\rho ^{2}\right]+2K\operatorname {Arc} .\operatorname {Tang} .={\frac {\rho \operatorname {Sin} .\omega -r\operatorname {Cos} .\theta }{\rho \operatorname {Cos} .\omega +r\operatorname {Sin} .\theta }}+\ldots \right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b842061bf96b26e21ed199416b07b807e77871f)
(51)
![{\displaystyle \int _{-\infty }^{+\infty }\operatorname {Arc} .\operatorname {Tang} .={\frac {s}{x}}.\varphi (x)\operatorname {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72c57a187800a4a5b63f181f28d5848fd4025398)
![{\displaystyle =\varpi \left\{H\operatorname {l} \left(1+2s\operatorname {Cos} .\omega +s^{2}\right)-2K\operatorname {Arc} .\operatorname {Tang} .={\frac {s\operatorname {Sin} .\omega }{\rho +s\operatorname {Cos} .\omega }}+\ldots \right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d4d8660569d0c1fc845b08bcf287dcd9cb7a715)
(52)
![{\displaystyle \int _{-\infty }^{+\infty }\varphi (x).\operatorname {Arc} .\operatorname {Cot} .x.\operatorname {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7376c8948cdfd66a0ded9ce22c45814656b31f7c)
![{\displaystyle =\varpi \left\{H\operatorname {l} \left(1+{\frac {2\operatorname {Cos} .\omega }{\rho }}+{\frac {1}{\rho ^{2}}}\right)-2K\operatorname {Arc} .\operatorname {Cot} .={\frac {\rho +\operatorname {Cos} .\omega }{\operatorname {Sin} .\omega }}+\ldots \right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6325aab3dccf55cc0038e588869befeb9c089d7)