Difference between revisions of "The Road to Reality Study Notes"

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:<math>\frac{n!}{2πi}\oint\frac{f(z)}{(z-p)^{n+1}}dz=f^{(n)}(p)</math>
:<math>\frac{n!}{2πi}\oint\frac{f(z)}{(z-p)^{n+1}}dz=f^{(n)}(p)</math>


If we use this to provide the definition of a derivative at a point, we can then construct a Maclaurin formula (if using the origin, otherwise the more general Taylor series) for f(z) using the derivatives in the coefficients of the terms.
If we use this to provide the definition of a derivative at a point, we can then construct a Maclaurin formula (if using the origin, otherwise the more general [https://en.wikipedia.org/wiki/Taylor_series Taylor series]) for f(z) using the derivatives in the coefficients of the terms.


:<math> \sum_{n=0} ^ {\infty} \frac {f^{(n)}(z)}{n!} (z-p)^{n}, </math>
:<math> \sum_{n=0} ^ {\infty} \frac {f^{(n)}(z)}{n!} (z-p)^{n}, </math>


This can be shown to sum to f(z), thereby showing the function has an actual nth derivative at the origin.  This concludes the argument showing that complex smoothness in a region surrounding the origin implies that the function is also holomorphic. Penrose notes that neither the premise (f(z) is complex-smooth) nor the conclusion (f(z) is analytic) contains contour integration or multivaluedness of a complex logarithm, yet these ingredients are essential for finding the route to the answer and that this is a ‘wonderful example of the way that mathematicians can often obtain their results’.
This can be shown to sum to $$f(z)$$, thereby showing the function has an actual $$n$$th derivative at the origin or general point p.  This concludes the argument showing that complex smoothness in a region surrounding the origin or point implies that the function is also holomorphic. Penrose notes that neither the premise ($$f(z)$$ is complex-smooth) nor the conclusion ($$f(z)$$ is analytic) contains contour integration or multivaluedness of a complex logarithm, yet these ingredients are essential for finding the route to the answer and that this is a ‘wonderful example of the way that mathematicians can often obtain their results’.


=== 7.4 Analytic continuation ===
=== 7.4 Analytic continuation ===


== Chapter 8 Riemann surfaces and complex mappings ==
== Chapter 8 Riemann surfaces and complex mappings ==
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