Page:Annales de mathématiques pures et appliquées, 1819-1820, Tome 10.djvu/227

219

DIVERSES.

${\begin{aligned}&\operatorname {Cos} .z={\frac {e^{z{\sqrt {-1}}}+e^{z{\sqrt {-1}}}}{2}},\\&\operatorname {Sin} .z={\frac {e^{z{\sqrt {-1}}}-e^{z{\sqrt {-1}}}}{2{\sqrt {-1}}}}\,;\end{aligned}}$

dans lesquelles on a, comme l’on sait

$e=1+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\ldots$

L’on en déduira

$z={\frac {1}{2{\sqrt {-1}}}}\operatorname {Log} .{\frac {\operatorname {Cos} .z+{\sqrt {-1}}\operatorname {Sin} .z}{\operatorname {Cos} .z-{\sqrt {-1}}\operatorname {Sin} .z}},\qquad$(2)

auquel on peut substituer

$z={\frac {1}{2{\sqrt {-1}}}}\operatorname {Log} .{\frac {\operatorname {Cos} .^{2}z+{\sqrt {-1}}\operatorname {Sin} .z\operatorname {Cos} .z}{\operatorname {Cos} .^{2}z-{\sqrt {-1}}\operatorname {Sin} .z\operatorname {Cos} .z}},$

ou encore

$z={\frac {1}{2{\sqrt {-1}}}}\operatorname {Log} .{\frac {1-\operatorname {Sin} .^{2}z+{\sqrt {-1}}\operatorname {Sin} .z\operatorname {Cos} .z}{1-\operatorname {Sin} .^{2}z-{\sqrt {-1}}\operatorname {Sin} .z\operatorname {Cos} .z}}\,;$

ou enfin

$z={\frac {1}{2{\sqrt {-1}}}}\operatorname {Log} .{\frac {1-\operatorname {Sin} .z\left(\operatorname {Sin} .z-{\sqrt {-1}}\operatorname {Cos} .z\right)}{1-\operatorname {Sin} .z\left(\operatorname {Sin} .z+{\sqrt {-1}}\operatorname {Cos} .z\right)}}\,;$

et par conséquent

$z={\frac {1}{2{\sqrt {-1}}}}\left\{\operatorname {Log} .\left[1-\operatorname {Sin} .z\left(\operatorname {Sin} .z-{\sqrt {-1}}\operatorname {Cos} .z\right)\right]\right.$

$\left.-\operatorname {Log} .\left[1-\operatorname {Sin} .z\left(\operatorname {Sin} .z+{\sqrt {-1}}\operatorname {Cos} .z\right)\right]\right\}.\quad$(3)

Faisons présentement

$\operatorname {Tang} .z=x,$

d’où