Most often, we use a power series for real values of its independent variable, but they work perfectly well for complex variables. It is actually easier to study the set of values for which a power series is valid, called its region or circle of convergence, if we think of it as a function of a complex variable. If a function has a power series expansion around some point $$a\text{,}$$ then the circle of convergence extends to the nearest point at which the function is not analytic. (Briefly, a function which is not analytic is singular in some way. A function is certainly not analytic at any point at which its value becomes infinite or at a branch point of a root.) For example, the function $$\frac{1}{z^2+1}$$ seems perfectly well-behaved if $$z$$ is real, but blows up if $$z=\pm i\text{.}$$ If we expand this function in a power series around $$z=0\text{,}$$ using the binomial expansion in SectionÂ 7.7, the resulting series is valid for $$\vert z\vert\le 1\text{,}$$ i.e. inside a circle with radius $$1\text{.}$$