c. a ter 5
e em -ntary
functions
Overview
How should cornplex-valued functions such as ez, log z, sin z, and the like, be
defined? Clearly, any responsible definition should satisfy the following criteria.
- The functions so defined must give the same values as the corresponding
functions for real variables when the number z is a real number. - As much as possible, the properties of these new functions must correspond
with their real counterparts. For example, we would want e•• +z^2 = e•• e"^2
to be valid regardless of whether z were real or complex.
These requirements rnay seem like a tall order to fill. There is a procedure,
however, that offers promising results. It is to put the expansion of the real
functions e", sinx, and so on, as power series in complex form. We use this
strategy in this chapter.
5.1 The Complex Exponential Function
Recall that the real exponential function can be represented by the power series
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e" = L: ~x". Thus, it is only natural to define the complex exponential e•,
n=O
also written as exp (z), in the following way.
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Definition 5.1: e' = exp(z) = 2: ;!iz".
n=O155