122
FACULTÉS
![{\displaystyle \operatorname {Log} .(p+iq)!=M+iN,\qquad \operatorname {Log} .(p-iq)!=M-iN.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea39d2ad7043867fbeacf60ea56c481b90f9a579)
Posant, de plus,
ce qui donne
![{\displaystyle k^{2}=1+q^{2},\qquad \operatorname {Tang} .\phi =q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f892595d773537a7741e5506f7aea657e7161f85)
![{\displaystyle M={\tfrac {1}{2}}\operatorname {Log} .k-q\phi -\Gamma 1+B_{2}.{\frac {\operatorname {Cos} .\phi }{k}}+B_{4}{\frac {\operatorname {Cos} .3\phi }{3k^{3}}}+\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8acb92dc44dce5c1db9972b25bbd3181ea233bf)
![{\displaystyle N=-q+{\tfrac {1}{2}}\phi +q\operatorname {Log} .k-B_{2}.{\frac {\operatorname {Sin} .\phi }{k}}-B_{4}.{\frac {\operatorname {Sin} .3\phi }{3k^{3}}}+\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f362fd80fd0afc60eb6e4ef6e245d694cabda99)
on aura
![{\displaystyle \operatorname {Log} .(+iq)!=M+iN,\qquad \operatorname {Log} .(-iq)!=M-iN,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f95d7b188cd8b2db34e9a6f310c9882de4be24e5)
ce qui donnera encore
![{\displaystyle \operatorname {Log} .(+iq)!(-iq)!=2M,\qquad \operatorname {Log} .{\frac {(+iq)!}{(-iq)!}}=-2iN.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f6908c0fed09fcf8ce783972df31da371c4ad8)
14. Le logarithme de
pouvant toujours être développé en une série de la forme
![{\displaystyle Ay+By^{2}+Cy^{3}+Dy^{4}+\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a28e4e36ac3ef9f35dced1cd712bfa5d40fd977)
on aura dans le cas actuel de ![{\displaystyle a=r=1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62ed9f76465780eadcbc62af83c64e707955ed5e)
![{\displaystyle {\begin{array}{ll}-\ \ A=\Lambda 1&=B_{1}+B_{2}+B_{4}+B_{6}+\ldots ,\\+2B=1+{\tfrac {1}{4}}+{\tfrac {1}{9}}+{\tfrac {1}{16}}+\ldots &={\tfrac {1}{1}}+{\tfrac {1}{2}}+2B_{2}+4B_{4}+6B_{6}+\ldots ,\\-3C=1+{\tfrac {1}{8}}+{\tfrac {1}{27}}+{\tfrac {1}{64}}+\ldots &={\tfrac {1}{2}}+{\tfrac {1}{2}}+3B_{2}+10B_{4}+21B_{6}+\ldots ,\\+4D=1+{\tfrac {1}{16}}+{\tfrac {1}{81}}+{\tfrac {1}{256}}+\ldots &={\tfrac {1}{3}}+{\tfrac {1}{2}}+4B_{2}+20B_{4}+56B_{6}+\ldots ,\\-5E=1+{\tfrac {1}{32}}+{\tfrac {1}{243}}+{\tfrac {1}{1024}}+\ldots &={\tfrac {1}{4}}+{\tfrac {1}{2}}+5B_{2}+35B_{4}+126B_{6}+\ldots ,\\+6F=1+{\tfrac {1}{64}}+{\tfrac {1}{729}}+{\tfrac {1}{4096}}+\ldots &={\tfrac {1}{5}}+{\tfrac {1}{2}}+6B_{2}+56B_{4}+252B_{6}+\ldots ,\\\ldots \ldots \ldots \ldots \ldots \ldots \ldots &\ldots \ldots \ldots \ldots \ldots \ldots \\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6afdfe53fd4585fa1af96f008af7cbf6b8919c26)
Les valeurs numériques de toutes ces sommes de puissances sont connues et calculées ; quant à celle de
elle est
![{\displaystyle 0,\ 57721\ 56649\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3248b991f6f7165e96b42eb700be26c02d09c9ab)
On sait de plus que les sommes à indice pair sont réductibles aux puissances paires de
; d’où l’on obtient