E465  Demonstratio theorematis Neutoniani de evolutione potestatum binomii pro casibus, quibus exponentes non sunt numeri integri
(A demonstration of a theorem of Newton on the expansion of the powers of a binomial by cases, in which the exponents are not integral numbers)
Summary:
In Calculus Differentialis, Euler showed that (1 + x)^{a} = 1 + C(a, 1)x + C(a, 2)x^{2} + .... . Here, Euler shows it to be true for arbitrary real values of a and makes use of the functional equation that if f(a) = (1 + x)^{a}, then f(a)f(b) = f(a + b).
According to the records, it was presented to the St. Petersburg Academy on July 1, 1773.
Publication:

Originally published in Novi Commentarii academiae scientiarum Petropolitanae 19, 1775, pp. 103111

Opera Omnia: Series 1, Volume 15, pp. 207  216
Documents Available:
 Original publication: E465
 German Translation (Alexander Aycock and Arseny Skryagin): E465
 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 E465 include:
 Dhombres J., “Some aspects of the history of functionalequations linked to the evolution of the function concept.” Archive for History of Exact Sciences, 36 (2), pp. 91181 (1986).
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