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Why study complex numbers, and functions of of a complex variable?

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 $x + i y$ .

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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