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The Road to Reality Study Notes: Difference between revisions
→7.1 Complex smoothness; holomorphic functions
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To do so, the notion of a special type of integration along a contour in the complex plane is to be defined. This integration can then be used to solve for the coefficients of a [https://en.wikipedia.org/wiki/Taylor_series Taylor series] expression which allow for us to see that any complex function which is complex-smooth in the complex plane is necessarily analytic, or holomorphic. | To do so, the notion of a special type of integration along a contour in the complex plane is to be defined. This integration can then be used to solve for the coefficients of a [https://en.wikipedia.org/wiki/Taylor_series Taylor series] expression which allow for us to see that any complex function which is complex-smooth in the complex plane is necessarily analytic, or holomorphic. | ||
As stated in 7.3, instead of directly providing the definition of | As will be stated in 7.3, instead of directly providing the definition of holomorphic functions, Penrose chooses to demonstrate the argument with the ingredients in order to show a "wonderful example of the way that mathematicians can often obtain their results. Neither the premise ($$f(z)$$ is complex-smooth) nor the conclusion ($$f(z)$$ is analytic) contains a hint of the notion of contour integration or the multivaluedness of a complex logarithm. Yet these ingredients provide the essential clues to the true route to finding the answer". | ||
=== 7.2 Contour integration === | === 7.2 Contour integration === | ||