part 2: intro to complex vbls:
In many ways the property of holomorphicity is analogous to that of being a polynomial, but more subtle. For example one is always interested, in mathematics, in the problem of existence and uniqueness of solutions to equations. For polynomials the fundamental theorem of algebra tells us that if f(z) is a complex polynomial, and if z0 is a complex number such that f(z) = z0 has no solutions, then f is a constant. The analogous existence theorem is this: if f:C-->C is a holomorphic function (defined and holomorphic everywhere), and if there are two complex numbers z0, and z1, such that neither of the equations f(z)= z0, nor f(z)= z1 has a solution, then f is a constant. The corresponding uniqueness theorems are these: if f,g are complex polynomials with exactly the same zeroes (including multiplicities), then the quotient f/g is a constant; and if f,g:C-->C are holomorphic functions with exactly the same zeroes, then f/g=eh , for some holomorphic function h:C-->C. Since ez is never zero, even for complex z, this is the best we could hope for.
There are other analogies with polynomials, as follows: if one is given a finite set {a1,...,an} of complex numbers, not necessarily distinct, then there exists a polynomial f(z) of degree n, whose set of zeroes is precisely this set, and such an f can be expressed as a (finite) product, one factor for each ai. Moreover, if f is a complex polynomial of degree n, and if f vanishes (i.e. has the value zero) at a set of n+1 distinct points, then f is the zero polynomial. Hence a polynomial of degree n is entirely known if its values are known on a given set of n+1 points. Moreover there is a formula, called Lagrange's interpolation formula, which expresses the value at any point in terms of a finite sum involving the values at the given n+1 points. [The generalization to holomorphic functions is perhaps not so obvious - can you guess it? For the second part you have to figure out what we would mean by a set of infinity plus one points!] In fact if {a1,...,an,...} is an infinite set of complex numbers, such that in any circle centered at the origin there are only a finite number of them, then there is a holomorphic function f:C-->C whose zeroes occur precisely at the given points (and with given multiplicities), and this holomorphic function can be expressed as an infinite product, with essentially one factor for each point. Moreover, if {a1,...,an,...} is a infinite sequence of distinct complex numbers which converges to a (finite) complex number a0, and if f:C-->C is an entire holomorphic function, (holomorphic functions defined on all of C are called entire), which vanishes on all the points ai in the sequence and hence also at a0, then f is identically zero. Consequently an entire holomorphic function is completely determined by its values on the points of a convergent sequence. So the notion of "infinity plus one" points turns out to mean an infinite sequence plus a limit point of that sequence. In particular, an entire holomorphic function is determined by its values on any circle, and indeed there is a formula, Cauchy's integral formula, which gives the values of the function at points inside the circle in terms of an integral over the values on the circumference of the circle!
The uniqueness property just stated tells us in particular that a differentiable function f:R-->R has at most one extension to a holomorphic function F:C-->C, whereas f always has infinitely many different extensions to a (R-) differentiable function G:R2-->R2. For example, sin(t), et, and a0+a1t+a2t2+...+antn, each has a unique extension to a holomorphic function on C, but 1/(1+t2), has no extension to a holomorphic function on all of C, and f(t)=exp(-1/t2) for t≠0 and f(0)=0, has no extension which is holomorphic on any neighborhood of 0, even though f(t) is infinitely differentiable on R. Now the function 1/(1+t2) does of course have a holomorphic extension to any open set in C that does not contain either of the points i or -i, and the extension is given by the same formula, 1/(1+z2). On the other hand there exist functions f:R-->R which have holomorphic extensions to certain open subsets of C, but such that the extensions may take several different values at the same point of C! That is, given a point z0 in C, there may be one holomorphic extension of f to one open set U containing z0 which has one value at z0, and there may be a second holomorphic extension of f to a second open set V which also contains z0 such that the second extension has a different value at z0 from the first one! For example, the real function f(t)=ln(t), defined and differentiable for t in R can be extended to a holomorphic function on the complement of any ray emanating from the origin in the complex plane. But if f1 denotes the holomorphic extension to the complement of the negative imaginary (y-) axis, and if f2 denotes the holomorphic extension to the complement of the positive imaginary axis, then f1(-1)=iπ, while f2(-1)=-iπ. So in cases of functions which extend only into part but not all of C, the values of the extension at a specific point are not determined by the point but rather by the whole open set on which the extension exists. It is true that for each connected open subset of C which meets R, there is at most one holomorphic extension of f to that set. So the holomorphic extension of a function is often inevitably multiple valued, and the attempt to work out the proper domain on which a full single-valued holomorphic extension can be defined led Riemann to the beautiful theory now called "Riemann surfaces". He analyzed the geometry of the resulting domains and thereby constructed the origins of the subject of topology. This problem was also studied abstractly by Abel and Galois in connection with the problem of integrating algebraic functions. [An algebraic function is one g(z) such that there exists a polynomial F(z,w) in two variables for which F(z,g(z))=0, for all z. g(z)=z1/2 is such a function since it satisfies this condition for F(z,w)=z-w2.] Indeed, Galois wrote down some of his very advanced insights on this question in the second part of the same famous letter which is so much better known for its statement of the criterion for a polynomial equation to be solvable in terms of radicals.
So much for motivation. Don't be dismayed if you don't follow all of this. Read it and think about it and refer back to it when the course has progressed further, and see how much more you will understand of it. Some of these results will not be proved until the second or third quarter. We will next take up a rapid review of the fundamental ideas of limits, continuity, completeness, compactness, and connectedness in the setting of arbitrary metric spaces. This will come in handy in this and every other course in which you need to construct solutions to problems by taking limits. Then we will go on to the geometry of the complex numbers and the Riemann sphere, elementary examples and properties of holomorphic functions, properties of power series, and then into the heart of complex analysis, the Cauchy theory of complex path integration. There is no logical necessity of waiting so long to go to the Cauchy theory, as you can see from the rapid introduction to it in the book of Knopp, but I think we will feel more comfortable with the calculus of complex functions if we get acquainted a bit with complex numbers and their topology first. Please let me know how the course is going for you, whether it is either too slow or too fast, too easy or too hard. I may not be able to solve all your problems but I can try if I know how things are going. And do lots of problems from Hille. To start with I want you to try all of them. We will see if this is feasible. In my opinion you will be very glad you did try them all when it comes time to be tested on the material.
Having said all this, I want to admit that I myself, and many others, think that complex analysis is worth studying simply because the results are so beautiful!
