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Functions of (x, y) that depend only on the combination (x
+ iy) are called functions of a complex variable and
functions of this kind that can be expanded in power series
in this variable are of particular interest.
This combination (x + iy) is generally called z, and we can
define such functions as zn, exp(z), sin z, and
all the standard functions of z as well as of x. They are
defined in exactly the same way the only difference being
that they are actually complex valued functions, that is,
they are vectors in this two dimensional complex number space,
each with a real and an imaginary part (or component). Most
of the standard functions we have previously discussed have
the property that their values are real when their arguments
are real. The obvious exception is the square root function,
which becomes imaginary for negative arguments.
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