Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/390

378

SOMMATION

 Les corrections sont expliquées en page de discussion

1.^o  $1+{\frac {a\operatorname {Cos} .t\operatorname {Cos} .u\operatorname {Cos} .v}{1}}+{\frac {a^{2}\operatorname {Cos} .2t\operatorname {Cos} .2u\operatorname {Cos} .2v}{1.2}}+{\frac {a^{3}\operatorname {Cos} .3t\operatorname {Cos} .3u\operatorname {Cos} .3v}{1.2.3}}+\ldots$

$=+{\frac {1}{4}}\left\{{\begin{aligned}&e^{a\operatorname {Cos} .(t+u+v)}.\operatorname {Cos} .\left[a\operatorname {Sin} .(t+u+v)\right]\\\\+&e^{a\operatorname {Cos} .(t+u-v)}.\operatorname {Cos} .\left[a\operatorname {Sin} .(t+u-v)\right]\\\\+&e^{a\operatorname {Cos} .(u+v-t)}.\operatorname {Cos} .\left[a\operatorname {Sin} .(u+v-t)\right]\\\\+&e^{a\operatorname {Cos} .(v+t-u)}.\operatorname {Cos} .\left[a\operatorname {Sin} .(v+t-u)\right]\end{aligned}}\right\}.$

2.^o  ${\frac {a\operatorname {Sin} .t\operatorname {Sin} .u\operatorname {Sin} .v}{1}}+{\frac {a^{2}\operatorname {Sin} .2t\operatorname {Sin} .2u\operatorname {Sin} .2v}{1.2}}+{\frac {a^{3}\operatorname {Sin} .3t\operatorname {Sin} .3u\operatorname {Sin} .3v}{1.2.3}}+\ldots$

$=-{\frac {1}{4}}\left\{{\begin{aligned}&e^{a\operatorname {Cos} .(t+u+v)}.\operatorname {Sin} .\left[a\operatorname {Sin} .(t+u+v)\right]\\\\-&e^{a\operatorname {Cos} .(t+u-v)}.\operatorname {Sin} .\left[a\operatorname {Sin} .(t+u-v)\right]\\\\-&e^{a\operatorname {Cos} .(u+v-t)}.\operatorname {Sin} .\left[a\operatorname {Sin} .(u+v-t)\right]\\\\-&e^{a\operatorname {Cos} .(v+t-u)}.\operatorname {Sin} .\left[a\operatorname {Sin} .(v+t-u)\right]\end{aligned}}\right\}.$

3.^o $1+{\frac {a\operatorname {Cos} .^{3}t}{1}}+{\frac {a^{2}\operatorname {Cos} .^{3}2t}{1.2}}+{\frac {a^{3}\operatorname {Cos} .^{3}3t}{1.2.3}}+\ldots$

$=+{\frac {1}{4}}\left\{e^{a\operatorname {Cos} .3t}.\operatorname {Cos} .(a\operatorname {Sin} .3t)+3e^{a\operatorname {Cos} .t}.\operatorname {Cos} .(a\operatorname {Sin} .t)\right\}.$

4.^o ${\frac {a\operatorname {Sin} .^{3}t}{1}}+{\frac {a^{2}\operatorname {Sin} .^{3}2t}{1.2}}+{\frac {a^{3}\operatorname {Sin} .^{3}3t}{1.2.3}}+\ldots$

$=-{\frac {1}{4}}\left\{e^{a\operatorname {Cos} .3t}.\operatorname {Sin} .(a\operatorname {Sin} .3t)-3e^{a\operatorname {Cos} .t}.\operatorname {Sin} .(a\operatorname {Sin} .t)\right\}.$

et ainsi de suite.