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Complex numbers provide an important example of a two dimensional
vector space; one in which vectors have additional structure:
they can be multiplied together as well as added.
This fact allows us to define most of our standard functions
that are ordinarily defined as functions of a real variable,
on the complex plane, where the variable is x+iy.
Such functions define a mapping from points on the plane to
one another, which functions have important and useful properties.
Moreover, we can define differentiation in the complex plane
and integration just as it is defined on the real line. Our
ability to create integrals along paths in the complex plane
gives us an amazingly powerful tool for evaluating integrals
and for solving classes of differential equations.
The relation between the distance from a point z' to the
nearest singularity of a function f and the radius of convergence
of the power series expansion of f about z (they are the same)
is a useful byproduct of the extension of definitions of functions
into the complex plane.
In applications complex valued functions of a complex variable
are very useful in describing waves and solutions to linear
differential equations that arise in physics. Quantum mechanics
could not be described at all without such functions.
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