D’où l’on déduira (18) la formule générale
où il faudra faire y = 1. {\displaystyle y=1.}
Soient, par exemple,
on trouvera
![{\displaystyle {\begin{aligned}\psi \left({\frac {bx}{a}}\right)&=\psi (y)\\&+a\ \ {\frac {\mathrm {A} }{b^{2}}}y^{2}\psi '(y)\\&+a^{2}\left[{\frac {\mathrm {B} }{b^{3}}}y^{3}\psi '(y)+{\frac {\mathrm {B} _{1}}{b^{4}}}.{\frac {1}{2}}{\frac {d\left[y^{4}\psi '(y)\right]}{dy}}\right]\\&+a^{3}\left[{\frac {\mathrm {C} }{b^{4}}}y^{4}\psi '(y)+{\frac {\mathrm {C} _{1}}{b^{5}}}.{\frac {1}{2}}{\frac {d\left[y^{5}\psi '(y)\right]}{dy}}+{\frac {\mathrm {C} _{2}}{b^{6}}}.{\frac {1}{2.3}}{\frac {d^{2}\left[y^{6}\psi '(y)\right]}{dy^{2}}}\right]\\&+a^{4}\left[{\frac {\mathrm {D} }{b^{5}}}y^{5}\psi '(y)+{\frac {\mathrm {D} _{1}}{b^{6}}}.{\frac {1}{2}}{\frac {d\left[y^{6}\psi '(y)\right]}{dy}}+{\frac {\mathrm {D} _{2}}{b^{7}}}.{\frac {1}{2.3}}{\frac {d^{2}\left[y^{7}\psi '(y)\right]}{dy^{2}}}\right.\\&\ \ \ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad +\left.{\frac {\mathrm {D} _{3}}{b^{8}}}.{\frac {1}{2.3.4}}{\frac {d^{3}\left[y^{8}\psi '(y)\right]}{dy^{3}}}\right]\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3c136505c54539593daa2520e7076abf1d3f267)
![{\displaystyle {\begin{aligned}{\frac {b^{m}x^{m}}{a^{m}}}=&1\\&+{\frac {m\mathrm {A} }{b}}{\frac {a}{b}}\\&+\left[{\frac {m\mathrm {B} }{b}}+{\frac {m(m+3)\mathrm {B} _{1}}{2b^{2}}}\right]{\frac {a^{2}}{b^{2}}}\\&+\left[{\frac {m\mathrm {C} }{b}}+{\frac {m(m+4)\mathrm {C} _{1}}{2b^{2}}}+{\frac {m(m+4)(m+5)\mathrm {C} _{2}}{2.3.b^{3}}}\right]{\frac {a^{3}}{b^{3}}}\\&+\left[{\frac {m\mathrm {D} }{b}}+{\frac {m(m+5)\mathrm {D} _{1}}{2b^{2}}}+{\frac {m(m+5)(m+6)\mathrm {D} _{2}}{2.3.b^{3}}}\right.\\&\quad \qquad \qquad \qquad \qquad \qquad +\left.{\frac {m(m+5)(m+6)(m+7)D_{3}}{2.3.4.b^{4}}}\right]{\frac {a^{4}}{b^{4}}}\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5eeb9b11225d59f099afe3450da1711c55241f6a)
