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
so that we have, for all real values of x,
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
- 2.
This chapter of Cauchy’s earlier text deals primarily with convergent imaginary series.
- 3.
The 1899 edition reads, “...the formula given in line 13 of page 192.” Cauchy is referencing the formula
$$\begin{aligned} \int _{0}^{\infty }{ z^ne^{-az} dz }=\frac{1\cdot 2\cdot 3\cdots n}{a^{n+1}}, \end{aligned}$$from his thirty-second lecture.
- 4.
The 1899 edition reads, “...the formula from line 14 of the cited page.” Here, Cauchy is refer-encing the formula
$$\begin{aligned} \int _{0}^{\infty }{ z^n e^{-az} \big (\cos {bz}+\sqrt{-1}\sin {bz}\big ) dz }=\frac{1\cdot 2\cdot 3\cdots n}{\big (a+b\sqrt{-1} \big )^{n+1}}, \end{aligned}$$from the same thirty-second lecture.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Cates, D.M. (2019). IMAGINARY EXPONENTIALS AND LOGARITHMS. USE OF THESE EXPONENTIALS AND OF THESE LOGARITHMS IN THE DETERMINATION OF DEFINITE OR INDEFINITE INTEGRALS.. In: Cauchy's Calcul Infinitésimal. Springer, Cham. https://doi.org/10.1007/978-3-030-11036-9_39
Download citation
DOI: https://doi.org/10.1007/978-3-030-11036-9_39
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-11035-2
Online ISBN: 978-3-030-11036-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)