Editing The Road to Reality Study Notes (section)

=== 7.3 Power series from complex smoothness ===
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 <math>p</math>) by what it is doing at a set of points surrounding the origin or the general point <math>p</math>. 

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

A 'higher-order' version of this formula allows us to inspect <math>n</math> number of derivatives with the same relationship.

:<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 [https://en.wikipedia.org/wiki/Taylor_series Taylor series]) for <math>f(z)</math> using the derivatives in the coefficients of the terms.

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

This can be shown to sum to <math>f(z)</math>, thereby showing the function has an actual <math>n</math>th derivative at the origin or general point <math>p</math>.  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 (<math>f(z)</math> is complex-smooth) nor the conclusion (<math>f(z)</math> 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’.