USE OF DERIVATIVES AND DIFFERENTIALS OF VARIOUS ORDERS IN THE EXPANSION OF ENTIRE FUNCTIONS

Dennis M. Cates ()

Chapter Chapter 19 in Cauchy's Calcul Infinitésimal, 2019, pp 97-102 from Springer

Abstract: 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 $$\begin{aligned} F(x)=a_0 + a_1x + a_2x^2 + \cdots + a_n x^n. \end{aligned}$$

Date: 2019
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DOI: 10.1007/978-3-030-11036-9_19

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