Œuvres de Charles Hermite publiées sous les auspices de l'Académie des sciences, par Émile Picard.

136 OEUVRES DE CHARLES 11lERI1TE. conduira t la fraction continue de Lambert: x tangx t a 11 ==X = - ----- 5-. et s'obtient ainsi. Soit A = sinx, puis successivement r ' A = A hx dx =sinx-x cosx, o A2= 0 A x dx = ( 3 — x2) sin- 3x cosx.R A3=f0 Aa clx = (5-6Xa2)sinx —(i5X —X3) coSX,..................................................... et, en general, A,+~i== o A,, xdx. Les formules elementaires f cosx F(x) dx sinx f (x) - cos '(), sinx F(x) dx = sinx ~'(x) -cosx f(x), o- l'on suppose F(x) un polynome entier et '(x) = F(x)- F"(x) -- Flv(x)-..., montrent que A,, est de la forme U sinx -+V cosx, U et V etant des polynomes entiers dont l'un est du degre n et l'autre du degre n -i. En second lieu, si l'on part du developpement en serie: A = sinx = x --- -- 2.3 2.3.4.5 " on en conclura aisement,,2/l-2+1 X 2n —3 I. 3.5... 2 7 -- I I2.3....5... 2 --?

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Title
Œuvres de Charles Hermite publiées sous les auspices de l'Académie des sciences, par Émile Picard.
Author
Hermite, Charles, 1822-1901.
Canvas
Page 130
Publication
Paris,: Gauthier-Villars,
1905-1917.
Subject terms
Mathematics.

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"Œuvres de Charles Hermite publiées sous les auspices de l'Académie des sciences, par Émile Picard." In the digital collection University of Michigan Historical Math Collection. https://name.umdl.umich.edu/aas7821.0003.001. University of Michigan Library Digital Collections. Accessed May 4, 2025.
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