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The Road to Reality Study Notes: Difference between revisions
→7.1 Complex smoothness; holomorphic functions
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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 == | ||