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).
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