22
INTÉGRATION
prenant enfin la demi-différence de ces deux-ci, on aura
![{\displaystyle 0=\left(Cz^{2}+12E\right)+Ez^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec1b0126c8de83961f3484c3e0c2d514b34d4219)
d’où, en remontant,
![{\displaystyle {\begin{aligned}0=&(Bz^{2}+6D)+Dz^{2},\\0=&(Az^{2}+2C).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11db5412c03729a78cfef2b1d449ce5f17e23231)
On tirera de ces trois dernières équations
![{\displaystyle {\frac {C}{z}}=-{\frac {1}{2}}A,\quad {\frac {D}{z^{3}}}=-{\frac {1}{6+z^{2}}}.{\frac {B}{z}},\quad {\frac {E}{z^{4}}}=+{\frac {A}{2\left(12+z^{2}\right)}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1e3377267151354e15e5e15716addb95730ac01)
les conditions relatives aux constantes sont d’ailleurs ici
![{\displaystyle 0=A-a{\frac {B}{z}}+a^{2}{\frac {C}{z^{2}}}-a^{3}{\frac {D}{z^{3}}}+a^{4}{\frac {E}{z^{4}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95b613658e62fa0aad5fd8c7aada86eabb4fc4de)
![{\displaystyle 1={\frac {B}{z}}-2a{\frac {C}{z^{2}}}+3a^{2}{\frac {D}{z^{3}}}-4a^{3}{\frac {E}{z^{4}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc5181fd9082bba49a1de2cf2c2e13f56321394)
en y substituant donc les trois valeurs ci-dessus, elles deviendront en faisant de suite
excepté dans le dénominateur de ![{\displaystyle B.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eccf5bca7cdc1fa4439af2d31831db6bde00473)
![{\displaystyle \left(1-{\frac {a^{2}}{2}}+{\frac {a^{4}}{24}}\right)A-\left(a-{\frac {a^{3}}{6}}\right){\frac {B}{z}}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f87c455a9e29e2526a1fde81d3ea1a84a38dc581)
![{\displaystyle \left(a-{\frac {a^{3}}{6}}\right)A+\left(1-{\frac {a^{2}}{2}}\right){\frac {B}{z}}=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9607fa57dfaada745e0c9176e22f5b1e3d58aca)
desquelles on tirera
![{\displaystyle A={\frac {a-{\frac {a^{3}}{6}}}{1-{\frac {a^{4}}{24}}+{\frac {a^{6}}{144}}}},\qquad {\frac {B}{z}}={\frac {1-{\frac {a^{2}}{2}}+{\frac {a^{4}}{24}}}{1-{\frac {a^{4}}{24}}+{\frac {a^{6}}{144}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2503dba731d5679d2aa59c00c2f6bc78d6a91ab)