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

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== Chapter 7 Complex-number calculus ==
== Chapter 7 Complex-number calculus ==
=== 7.1 Complex smoothness; holomorphic functions ===
=== 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 holomorphic functions, which play a vital role in much of the mathematical as well as physics material later in the book.
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 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.
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.


=== 7.2 Contour integration ===
=== 7.2 Contour integration ===


== Chapter 8 Riemann surfaces and complex mappings ==
== Chapter 8 Riemann surfaces and complex mappings ==
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