Abstract
It is easy to expand an entire function of x into a polynomial ordered according to the ascending powers of this variable, when we know the particular values of the function and of its successive derivatives, for \(x=0.\) In fact, denote by F(x) the given function, by n the degree of this function, and by \( a_0, a_1, a_2, \dots , a_n \) the unknown coefficients of the various powers of x in the expansion we seek, so that we have
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Notes
- 1.
What has developed here so simply is a special case of a Maclaurin series, one in which the function is exactly represented by a finite polynomial.
- 2.
The known formula Cauchy is referring to is the binomial expansion for the case n a positive integer.
- 3.
This note turns out to be a fairly long aside. However, this tangent is a precursor to important work to follow. To provide motivation, recall Cauchy is writing this text in the early 1820s, a time period in which many mathematicians were still following Lagrange’s earlier work on the calculus. The Taylor series is central to Lagrange, as the derivative of the function f(x) in Lagrange’s calculus is defined as the coefficient of the linear term in the Taylor series expansion of that function. Hence, the Taylor series was certainly a prominent topic for most of Cauchy’s contemporaries. Throughout this 1823 text, Cauchy fully develops his analysis of the Taylor and Maclaurin series, ending the book with examples clearly demonstrating the failure of Lagrange’s theory. This current Note is part of this pivotal complete development.
- 4.
In Lecture Six, Cauchy develops a technique to deal with this indeterminate situation in some cases. Within that lecture he shows in the limit, the ratio \(\frac{z}{y}\) approaches the same value as the ratio \(\frac{\varDelta z}{\varDelta y}, \) or \(\frac{dz}{dy}, \) which has also been shown to equal \(\frac{z^{\prime }}{y^{\prime }}\) at the point of interest. However, in the present case, with \(m>1\) as Cauchy has supposed, \(z^{\prime }\) and \(y^{\prime }\) are both still zero at \(x=a, \) and so, the technique fails as this second fraction is still an indeterminate form of \(\frac{0}{0}.\)
- 5.
The expressions here have been corrected. In both of Cauchy’s texts, y and z terms occur at the beginning of the right-hand side in each of these two equations, respectively, but they should not be present.
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Cates, D.M. (2019). USE OF DERIVATIVES AND DIFFERENTIALS OF VARIOUS ORDERS IN THE EXPANSION OF ENTIRE FUNCTIONS.. In: Cauchy's Calcul Infinitésimal. Springer, Cham. https://doi.org/10.1007/978-3-030-11036-9_19
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