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The Road to Reality Study Notes: Difference between revisions
→7.3 Power series from complex smoothness
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=== 7.3 Power series from complex smoothness === | === 7.3 Power series from complex smoothness === | ||
The example in section 7p2 is a particular case for the well-known Cauchy Formula, which allows us to know what the function is doing at the origin (or another general point p) by what it is doing at a set of points surrounding the origin (or the general point p). <math display="block">\frac{1}{2πi}\oint\frac{f(z)}{z-p}dz=f(p)</math> | |||
A higher-order version of this formula allows us to inspect n number of derivatives with the same relationship. | |||
If we use this to provide the definition of a derivative at a point, then we can construct a Maclaurin formula f(z) using the derivatives in the coefficients of the terms. | |||
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’. | |||
=== 7.4 Analytic continuation === | === 7.4 Analytic continuation === | ||
== Chapter 8 Riemann surfaces and complex mappings == | == Chapter 8 Riemann surfaces and complex mappings == | ||