Prerequisites: Required, Math 324 (multivariable calculus) and Math 327 (basic knowledge of limits and continuity, sequences and series); recommended, Math 328. It will also be helpful to have taken Math 308 (basic linear algebra).
Note to students from the honors sequence 334/335/336: You cannot get credit for both 336 and 427, but you can get credit for both 336 and 428. If you took 334 and/or 335 but NOT 336 and want to take 427, please see me to discuss whether this is advisable and possible to do for credit.
General teaching style: interactive lecturing, some individual and group work during class.
Complex analysis plays a substantial role in both theoretical and applied mathematics. The seemingly minor change from doing calculus with real numbers to using complex numbers (that is, including i = square root of -1) produces powerful consequences. A single complex number z can be written as a pair of real numbers: z = x+iy = (x,y). Thus the complex numbers live on a plane (as compared to the real numbers, which live on the number line), introducing a much richer geometry. The set of complex numbers can be manipulated algebraically in much the same way as the real numbers. When calculus is added to this mix of geometry and algebra, the interplay of the three produces results and techniques that have both intrinsic mathematical beauty and great power in applications.
After exploring some basic properties of complex numbers (Chapter 1), differentiation of functions with complex domain and range (Chapter 2), and the elementary functions as mappings of the complex plane (Chapter 3), we'll study several topics that reappear from real variable calculus, but with more powerful results. The geometry of the complex plane will explain for instance where power series converge and diverge. Our study of complex integration will culminate with "contour integrals." (Results obtained using contour integrals show up in various areas of mathematics and in such applied fields as statistics, physics, and engineering.) For both power series and integration, the properties of complex functions will allow us to draw conclusions from very limited information. For example, we can use the behavior of a function near only a few points to determine large scale results. This work in Chapters 4, 5, 6, and 7 will fill the rest of fall quarter.
In winter quarter, we will finish any remaining topics from Chapters 6 and 7, and revisit the elementary functions, this time with even more emphasis on geometry (Chapter 8). The combined geometric and analytic (i.e. calculus) properties of these and other complex functions imply that the functions are "conformal maps" (Chapter 9). As applications, we'll solve problems about fluid flows, temperature distributions, and electrostatic fields. (Chapters 10-12; no background in these applications will be assumed.) Key features of our solutions will have both mathematical and physical interpretations. As time allows, we will investigate additional topics.
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Most recently updated on January 2, 2013