3 Functions. Limits and Continuity. Arcs and Curves
3.1 Complex Functions
Mappings and, in particular, functions have been defined in Section 0.2.
Functions from the complex plane, or from a subset of the complex plane,
into the complex plane, are called complex functions. We have already
encountered examples of such functions and others will be defined in the
sequel. Examples of single-valued complex functions are
I(z) = z J(z) = z, w = lzl,
Examples of multiple-valued complex functions are"
W=*-JZ, w = argz, w = logz
the last two being defined on C - {O}.
At times it is convenient to specify that either the domain or the rang~
of a complex function is real. Thus we refer to w = lz I as a real function
of a complex variable (more specifically, as a function from C onto JR+),
while w = t + it^2 , 0 .:S t .:S 2, defines a complex function of a real variable
(more specifically, a function frorri [0,2] into C).
Real functions of a real variable are also considered as special cases of
complex functions of a complex variable. As usual, by a "variable" we
mean a symbol representing an undetermined element of a certain set.
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