5
Elementary Functions
In this chapter we discuss some elementary functions of a complex variable,
namely certain of the most simple algebraic functions as well as the so-
called elementary transcendental functions: the exponential, circular, and
hyperbolic functions, together with their inverses.
Some of these functions have been defined and used in preceding chap-
ters, but here the emphasis will be on the algebraic properties of such
functions and on the geometric aspects of the mapping defined by them.
With the exception of the inverses of the linear and bilinear functions, all
other inverse functions are multi-valued. To discuss those inverse functions
in a most intuitive (geometric) form, the introduction of the corresponding
Riemann surfaces will be made.
5.1 The Translation w = z + b
From the geometric interpretation of the sum of two complex numbers, it
follows that the mapping defined by the function w = z + b (b a constant),
regarded as a mapping of the complex plane C into itself, amounts to the
translation of each point z in the magnitude and direction of the vector
b (Fig. 5.1).
As a consequence, each geometric figure in the plane will be mapped by
this function into a congruent figure. In particular, straight lines will be
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