Analysis I pp 325-423 | Cite as

Powers, Exponentials, Logarithms, Trigonometric Functions

  • Roger Godement
Part of the Universitext book series (UTX)


The general theorems of Chapters II and III have allowed us, in passing, to establish most of the principal properties of the elementary functions which crop up everywhere in analysis. In this chapter we shall go over it all systematically. One can do this in various ways, each as instructive as the other. A first method (no 1 to 8) consist s of erecting the theory from a minimum of knowledge, in particular using neither the theory of power series nor the function exp, particularly its addition formula. A second, contrasting, method, starts from these and goes much further. In particular it allows us to construct a rigorous analytic theory of the trigonometric functions. A third method would be to define the function log x as an integral and from this deduce its properties as well as those of the exponential functions.


Power Series Series Expansion Trigonometric Function Convergent Series Infinite Product 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Roger Godement
    • 1
  1. 1.Département de MathématiquesUniversité Paris VIIParis Cedex 05France

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