| Complex Analysis I | Bent Petersen |
The prerequisites for complex analysis are difficult to list -- a bit of everything is useful. For the first quarter complex analysis one should know something about the most of the following topics:
While complex numbers arose quite naturally in many areas it was not until 1799 and later that Wessel, Argand and Gauss popularized the geometric description of complex numbers and operations on them in terms of points in the plane. By 1831 Gauss had fully accepted complex numbers as entities no more mysterious than real numbers. (Of course, after studying real analysis for any length of time most thoughtful people are prepared to admit that real numbers are pretty mysterious.)
You should review the basic of complex numbers through the introduction of polar coordinates in the complex plane. It may be useful too to be familiar with the complex exponential though we will of course study it.
Some facility in manipulating complex numbers will be assumed. Geometric considerations such as descriptions of lines and half-planes is not important to master now. Leave it for later.
You should review a bit of metric space theory. The notions of convergence, completeness and compactness are critical. In Mth 514 it is likely that the only metric spaces we will use seriously are the plane, the sphere, and perhaps the torus. Near the end of Mth 514 or in Mth 515 however we may make serious use of function spaces. The value of the abstract framework will then become obvious.
The extended complex plane (Riemann sphere) and the stereographic projection of the sphere onto the extended plane will be covered quickly in Mth 514. You should take some time to work through the calculations. You should convince yourself that the standard metric in the plane and the chordal metric restricted to the plane are equivalent on any bounded subset of the plane, but not otherwise (and you should understand what that means).
Möbius transformations (fractional linear transformations) will be discussed lightly in Mth 514. They are best viewed as homeomorphisms of the sphere. They are very important, but we will not make much use of them. Mostly we need them because each automorphism of the unit disk is a Möbius transform (of a particular type). You will not need to master a great deal of information about these transforms but you may find it amusing (and challenging) to investigate their mapping properties -- a great deal is known. In class we will just have an unreasonably fast introduction. Whenever we need to make use of Möbius transformations we will summarize what we need.
For the most part we will deal with power series, or series that are close to power series. You probably have learned quite a bit about power series, but we will re-derive all of it (in a complex context). We may occasionally refer also to Fourier series, but we only use the simplest results.
We will prove the Fundamental Theorem of Algebra so in sense you do not need to know it. It would be best if it is not a total surprise though.
At some point we will derive the partial fraction decomposition of a rational function, so once again you do not really need to know it. Of course, it will be much more illuminating if you have struggled with partial fractions in the past.
You should know a little bit about linear algebra. Not much is needed beyond the notion of a vector space and linear maps, and change of field.
It may be useful to know a little topology, especially the notion of homotopy of curves. While this particular concept will be explained, you will probably be more comfortable with it if you have encountered it before.
If you do find yourself at a loss in some area let me know. I will try to bring you up to speed or to suggest some reading. Much of what we will do is very beautiful, remarkably useful, and not too difficult to master. It would be a shame to let some small gap in your knowledge frustrate your studies.