E750 -- Commentatio in fractionem continuam, qua illustris La Grange potestates binomiales expressit
(A commentary on the continued fraction by which the illustrious La Grange has expressed the binomial powers)
Summary:
Euler gives a continued fraction representation of (1 + x)n.
involving 1,3,5,7,... and n^2-1,n^2-4,n^3-9,...
and squares of z, for x=2y and y=z/(1-z). He evaluates this continued fraction at z=t sqrt(-1), for ``vanishing''
n, and for infinite n.
Publication:
-
riginally published in Mémoires de l'académie des sciences de St.-Petersbourg 6, 1818, pp. 3-11
-
Opera Omnia: Series 1, Volume 16, pp. 232 - 240
Documents Available:
- Original Publication: E750
- English translation (Jordan Bell): E750
- German translation (Artur Diener and Alexander Aycock): E750
- The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E750 include:
- Ku YH., “Solution of Riccati equation by continued fractions.” Journal of the Franklin Institute-Engineering and Applied Mathematics, 293 (1), pp. 59-& (1972).
- Eric Weisstein's definition of natural logarithm, available here.
Return to the Euler Archive