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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 | 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. | ||
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 | 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. | ||
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". | 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". |