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The Road to Reality Study Notes: Difference between revisions
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→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). | The example in section 7p2 is a particular case for the well-known [https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula Cauchy Integral 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>\frac{1}{2πi}\oint\frac{f(z)}{z-p}dz=f(p)</math> | :<math>\frac{1}{2πi}\oint\frac{f(z)}{z-p}dz=f(p)</math> | ||
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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, | 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. | ||
:<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 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’. | ||