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

  • Dennis M. CatesEmail author
Chapter

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 always 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
$$\begin{aligned} A^x=1+\frac{x \, \varvec{l}A}{1}+\frac{x^2(\varvec{l}A)^2}{1\cdot 2}+\frac{x^3(\varvec{l}A)^3}{1\cdot 2\cdot 3}+\cdots . \end{aligned}$$

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Sun CityUSA

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