Editing The Road to Reality Study Notes (section)

=== 7.4 Analytic continuation ===
We now know that complex smoothness throughout a region is equivalent to the existence of a power series expansion about any point in the region.  A region here is defined as a open region, where the boundary is not included in the domain.

For example, if there is no singularity in the function, the region can be thought of as a circle of infinite radius. Taking <math>f(z)=\frac{1}{z}</math> however forces an infinite number of circles centered at any point with boundary radii passing through the origin (noting that an open region does not contain the boundary) to construct the domain.

Now we consider the question, given a function <math>f(z)</math> holomorphic in domain <math>D</math>, can we extend the domain to a larger <math>D’</math> so that <math>f(z)</math> also extends holomorphically?  A procedure is formed in which we use a succession of power series about a sequence of points, forming a path where the circles of convergence overlap.  This then results in a function that is uniquely determined by the values in the initial region as well as the path along which it was continued.  Penrose notes this [https://en.wikipedia.org/wiki/Analytic_continuation analytic continuation] as a remarkable ‘rigidity’ about holomorphic functions.

An example of this rigidity and path dependence is ‘our old friend’ <math>logz</math>.  There is no power series expansion about the origin due to the singularity there but depending on the path chosen of points around the origin (clockwise or anti-clockwise) the function extends or subtracts in value by <math>2πi</math>.  See chapter 5 and the euler formula (<math>e^{πi}=-1</math>) for a refresher.