Page:Annales de mathématiques pures et appliquées, 1812-1813, Tome 3.djvu/124

120

FACULTÉS

$a+pr+iqr=k(\operatorname {Cos} .\phi +i\operatorname {Sin} .\phi ),$

$\operatorname {Log} .(a+pr+iqr)=\operatorname {Log} .k+i\phi ,$

le logarithme de $a^{p+iq|r}$ prendra donc la forme d’un binôme $M+iN,$ dans lequel on aura

$M=p(-1+\operatorname {Log} .a)+\left(p-{\frac {1}{2}}+{\frac {a}{r}}\right)\operatorname {Log} .{\frac {k}{a}}-q\phi -\Gamma {\frac {r}{a}}$

$+B_{2}.{\frac {r}{k}}\operatorname {Cos} .\phi +B_{4}.{\frac {r^{3}}{3k^{3}}}\operatorname {Cos} .3\phi +B_{6}.{\frac {r^{5}}{5k^{5}}}\operatorname {Cos} .5\phi +\ldots ,$

$N=q(-1+\operatorname {Log} .a)+\left(p-{\frac {1}{2}}+{\frac {a}{r}}\right)\phi +q\operatorname {Log} .{\frac {k}{a}}$

$-B_{2}.{\frac {r}{k}}\operatorname {Sin} .\phi -B_{4}.{\frac {r^{3}}{3k^{3}}}\operatorname {Sin} .3\phi -B_{6}.{\frac {r^{5}}{5k^{5}}}\operatorname {Sin} .5\phi -\ldots \,;$

les deux séries sont convergentes à volonté.

11. Ayant ainsi

$\operatorname {Log} .\left(a^{p+iq|r}\right)=M+iN,$

il est visible qu’on aura

$\operatorname {Log} .\left(a^{p-iq|r}\right)=M-iN,$

ce qui donnera

$\operatorname {Log} .\left(a^{p+iq|r}\right)\left(a^{p-iq|r}\right)=2M,$

$\operatorname {Log} .{\frac {a^{p+iq|r}}{a^{p-iq|r}}}=2iN\,;$

d’où on conclura

${\frac {a^{p+iq|r}}{a^{p-iq|ra}}}=e^{2iN}=\operatorname {Cos} .2N+i\operatorname {Sin} .2N.$

Dans le cas, très-fréquent, de $p=0,$ lequel donne

$k^{2}=a^{2}+q^{2}r^{2},\qquad \operatorname {Tang} .\phi ={\frac {qr}{a}}\,;$

on aura

$M=\left({\tfrac {a}{r}}-{\tfrac {1}{2}}\right)\operatorname {Log} .{\tfrac {k}{a}}-q\phi -\Gamma {\tfrac {r}{a}}+B_{2}.{\tfrac {r}{k}}\operatorname {Cos} .\phi +B_{4}.{\tfrac {r^{3}}{3k^{3}}}\operatorname {Cos} .3\phi +\ldots ,$

$N=q(-1+\operatorname {Log} .a)+\left({\tfrac {a}{r}}-{\tfrac {1}{2}}\right)\phi +q\operatorname {Log} .{\tfrac {k}{a}}-B_{2}.{\tfrac {r}{k}}\operatorname {Sin} .\phi -B_{4}.{\tfrac {r^{3}}{3k^{3}}}\operatorname {Sin} .3\phi -\ldots \,;$

et par suite

$\operatorname {Log} .a^{iq|r}=M+iN,$

$\operatorname {Log} .a^{-iq|r}=M-iN,$