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

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.

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.

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’.

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’.