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The Road to Reality Study Notes: Difference between revisions
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→7.4 Analytic continuation
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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}< | 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 $$f(z)$$ holomorphic in domain $$D$$, can we extend the domain to a larger $$D’$$ so that $$f(z)$$ 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. | Now we consider the question, given a function $$f(z)$$ holomorphic in domain $$D$$, can we extend the domain to a larger $$D’$$ so that $$f(z)$$ 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’ $$logz$$. 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 $$2πi$$. See chapter 5 and the euler formula (<math>e^{πi}=-1< | An example of this rigidity and path dependence is ‘our old friend’ $$logz$$. 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 $$2πi$$. See chapter 5 and the euler formula (<math>e^{πi}=-1</math>) for a refresher. | ||
== Chapter 8 Riemann surfaces and complex mappings == | == Chapter 8 Riemann surfaces and complex mappings == | ||