Editing The Road to Reality Study Notes
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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. | 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 | 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 that pass 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>Dâ</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 ârigidityâ about holomorphic functions. | 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>Dâ</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 ârigidityâ about holomorphic functions. |