IMAGINARY EXPONENTIALS AND LOGARITHMS. USE OF THESE EXPONENTIALS AND OF THESE LOGARITHMS IN THE DETERMINATION OF DEFINITE OR INDEFINITE INTEGRALS

Dennis M. Cates ()

Chapter Chapter 39 in Cauchy's Calcul Infinitésimal, 2019, pp 215-219 from Springer

Abstract: Abstract We have proven in the thirty-seventh lecture that the exponential $$A^x$$ (A denoting a positive constant, and x a real variable) is alwaysImaginary logarithm Imaginary exponential equivalent to the sum of the series $$\begin{aligned} 1, \ \ \ \ \ \frac{x \, \varvec{l}A}{1}, \ \ \ \ \ \frac{x^2(\varvec{l}A)^2}{1\cdot 2}, \ \ \ \ \ \frac{x^3(\varvec{l}A)^3}{1\cdot 2\cdot 3}, \ \ \ \ \ \dots , \end{aligned}$$ so that we have, for all real values of x, $$\begin{aligned} A^x=1+\frac{x \, \varvec{l}A}{1}+\frac{x^2(\varvec{l}A)^2}{1\cdot 2}+\frac{x^3(\varvec{l}A)^3}{1\cdot 2\cdot 3}+\cdots . \end{aligned}$$

Date: 2019
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DOI: 10.1007/978-3-030-11036-9_39

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