77
DES FONCTIONS CIRCULAIRES.
![{\displaystyle {\tfrac {m}{1}}.{\tfrac {m-1}{2}}\int z^{n}\operatorname {d} z.\operatorname {Cos} .(m-4)z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cf38077b9ac7f1c003f7faf65229ca2f827bad2)
![{\displaystyle \left.-\ldots \mp {\tfrac {m}{1}}.{\tfrac {m-1}{2}}\ldots {\tfrac {{\tfrac {1}{2}}m+2}{{\tfrac {1}{2}}m-1}}\int z^{n}\operatorname {d} z.\operatorname {Cos} .2z\right\}+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84e05dffa72e4ce78965c9014beaa0120db2c9e3)
![{\displaystyle {\tfrac {1}{2}}.{\tfrac {3}{4}}\ldots {\tfrac {m-1}{m}}.{\tfrac {1}{n+1}}z^{n+1}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/faa8347a5e733971d898be295ab98cfaf8b7874b)
et, pour le cas de
nombre positif et impair,
(f)
![{\displaystyle \int z^{n}\operatorname {d} z\operatorname {Sin} .^{m}z={\tfrac {1}{2^{m-1}}}\left\{\int z^{n}\operatorname {d} z.\operatorname {Sin} .mz-{\tfrac {m}{1}}\int z^{n}\operatorname {d} z.\operatorname {Sin} .(m-2)z+\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce62ba949d5ac7384352db190e9fbecd631c8ae3)
![{\displaystyle {\tfrac {m}{1}}.{\tfrac {m-1}{2}}\int z^{n}\operatorname {d} z.\operatorname {Sin} .(m-4)z-\ldots \mp {\tfrac {m}{1}}.{\tfrac {m-1}{2}}\ldots {\tfrac {{\tfrac {1}{2}}m+5}{{\tfrac {1}{2}}m-3}}\int z^{n}\operatorname {d} z.\operatorname {Sin} .3z\pm }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2b48c9878b5ef72c451c6d00889b33df216c426)
![{\displaystyle \left.{\tfrac {m}{1}}.{\tfrac {m-1}{2}}\ldots {\tfrac {{\tfrac {1}{2}}m+3}{{\tfrac {1}{2}}m-1}}\int z^{n}\operatorname {d} z.\operatorname {Sin} .z\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bba4a014d5be830f9f4bfa5542544b33765d3e4)
Multipliant et divisant chacun des termes des seconds membres de ces deux équations affectés du signe d’intégration, par la
puissance du coefficient de
sous le signe de cosinus, équation (e), et sous celui de sinus, équation (f), en remarquant qu’en général ![{\displaystyle k\operatorname {d} z\operatorname {Cos} .kz=\operatorname {d} \operatorname {Sin} .kz,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03b2687ff8cac25b3f32e657f20b9d34a8e374bc)
on trouvera, pour le cas de
nombre positif et pair,
(5)
![{\displaystyle \quad \int z^{n}\operatorname {d} z\operatorname {Sin} .^{m}z=\pm {\tfrac {1}{2^{m-1}}}\left\{{\tfrac {1}{m^{n+1}}}\int (mz)^{n}\operatorname {d} .\operatorname {Sin} .(mz)\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e045a73ae2ad1cd87b9665231235aafaf8626035)
![{\displaystyle -{\tfrac {m}{(m-2)^{n+1}}}\int \left[(m-2)z\right]^{n}\operatorname {d} .\operatorname {Sin} .\left[(m-2)z\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcee10c4118c0cd195aa89300ef023f084c7f807)
![{\displaystyle +{\tfrac {m}{1}}.{\tfrac {m-1}{2}}.{\tfrac {1}{(m-4)^{n+1}}}\int \left[(m-4)z\right]^{n}\operatorname {d} .\operatorname {Sin} .\left[(m-4)z\right]\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9844b91bdf1497567d1ce0892352accf0668c09f)
![{\displaystyle \left.\mp {\tfrac {m}{1}}.{\tfrac {m-1}{2}}\ldots {\tfrac {{\tfrac {1}{2}}m+2}{{\tfrac {1}{2}}m-1}}.{\tfrac {1}{2^{n+1}}}\int (2z)^{n}\operatorname {d} .\operatorname {Sin} .(2z)\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d54b3d7118e861f172434cf792738e8c6c8ac274)
![{\displaystyle +{\tfrac {1}{2}}.{\tfrac {3}{4}}\ldots {\tfrac {m}{m-1}}.{\tfrac {n+1}{1}}z^{n+1}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e7c82bbfeda086061da7f1a3cf13e76f8e060e6)
les signes supérieurs devant être pris lorsque
est un nombre doublement pair et les inférieurs dans le cas contraire.