Page:Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/103

(53)$\int _{-\infty }^{+\infty }e^{a\operatorname {Cos} .bx}.\operatorname {Cos} .(a\operatorname {Sin} .bx).\varphi (x).\operatorname {d} x$

$=-2\varpi \left\{e^{ae^{-bk}.\operatorname {Cos} .bh}\left[K\operatorname {Cos} .\left(ae^{-bx}.\operatorname {Sin} .bh\right)+H\operatorname {Sin} .\left(ae^{-bx}.\operatorname {Sin} .bh\right)\right]+\ldots \right\}.$

(54)$\int _{-\infty }^{+\infty }e^{a\operatorname {Cos} .bx}.\operatorname {Sin} .(a\operatorname {Sin} .bx).\varphi (x).\operatorname {d} x$

$=2\varpi \left\{e^{ae^{-bk}.\operatorname {Cos} .bh}\left[H\operatorname {Cos} .\left(ae^{-bx}.\operatorname {Sin} .bh\right)-K\operatorname {Sin} .\left(ae^{-bx}.\operatorname {Sin} .bh\right)\right]+\ldots \right\}.$

(55)$\int _{-\infty }^{+\infty }\operatorname {l} \left(1+2r\operatorname {Cos} .bx+r^{2}\right)\varphi (x)\operatorname {d} x$

$=-2\varpi \left\{K\operatorname {l} \left(1+2re^{-bk}.\operatorname {Cos} .bh+r^{2}e^{-2bk}\right)+2H\operatorname {Arc} .\operatorname {Tang} .={\frac {r\operatorname {Sin} .bh}{e^{bk}+r\operatorname {Cos} .bh}}+\ldots \right\}.$

(56)$\int _{-\infty }^{+\infty }\operatorname {Arc} .\operatorname {Tang} .={\frac {r\operatorname {Sin} .bx}{1+r\operatorname {Cos} .bx}}.\varphi (x)\operatorname {d} x$

$=\varpi \left\{H\operatorname {l} \left(1+2re^{-bk}.\operatorname {Cos} .bh+r^{2}e^{-2bk}\right)-2K\operatorname {Arc} .\operatorname {Tang} .={\frac {r\operatorname {Sin} .bh}{e^{bk}+r\operatorname {Cos} .bh}}+\ldots \right\}.$

On trouvera encore, en prenant pour $\varphi (x)$ une fonction paire de la variable $x$,

(57)$\int _{-\infty }^{+\infty }x^{a-1}\operatorname {Sin} .\left({\frac {a\pi }{2}}-bx\right).\varphi (x)\operatorname {d} x$

$=-\varpi \left\{\rho ^{a-1}e^{-bk}\left[K\operatorname {Cos} .(\omega -a\omega +bh)+H\operatorname {Sin} .(\omega -a\omega +bh)\right]+\ldots \right\}.$

Il est facile de reconnaître les différentiations que doivent subir les diverses formules auxquelles nous sommes parvenus, dans