Editing The Road to Reality Study Notes (section)

=== 7.1 Complex smoothness; holomorphic functions ===
An outline for section 7 is presented, in which calculus with complex numbers is introduced.  The material in this chapter leads up to the explanation of [https://en.wikipedia.org/wiki/Holomorphic_function holomorphic functions], which play a vital role in much of the mathematical as well as physics material later in the book.

To do so, the notion of a special type of integration along a contour in the complex plane is to be defined. This integration can then be used to solve for the coefficients of a [https://en.wikipedia.org/wiki/Taylor_series Taylor series] expression which allow for us to see that any complex function which is complex-smooth in the complex plane is necessarily analytic, or holomorphic.

As will be stated in 7.3, instead of directly providing the definition of holomorphic functions, Penrose chooses to demonstrate the argument with the ingredients in order to show a "wonderful example of the way that mathematicians can often obtain their results. Neither the premise (<math>f(z)</math> is complex-smooth) nor the conclusion (<math>f(z)</math> is analytic) contains a hint of the notion of contour integration or the multivaluedness of a complex logarithm.  Yet these ingredients provide the essential clues to the true route to finding the answer".