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

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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, then we can construct a Maclaurin formula 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 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’.
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