here is a brief course in complex variables, aka complex analysis:
1) unlike real variables, if a function of a complex variable has even one derivative, at every point of an open domain, then it has infinitely many derivatives, and even better, it is represented by a convergent power series in any circle contained in that domain. Thus differenmtiable complex functions are called "analytic" or sometimes holomorphic.
2) since all power series enjoy the principle of "isolated zeroes", so also do all differentiable complex functions. i.e. if a complex analytic function in a connected domain D, equals zero on a set of points which has an accumulation point in D, then the function is dientically zero in D.
3) cor: every real analytic function, such as sin, cos, e^x, etc etc, has at most one analytic extension to the comple plane, and in afct these all do have an extension.
4) indeed any analytic real functions does have an extension to some open domain in the plane, since the same series expansion that converges on an interval of radius r on the real line, also converges in the disc of radius r in the plane (and same center).
this explains why the innocent loking function 1/(1+x^2), which is infinitely differentiable on the real line, ahs a taylor series that only converges in the unit interval. i.e. it cannot converge at z=i, at distance 1 from the origin, so it cannot converge on any interval of radius larger than 1.
5) since a complex function define even in a tiny open disc, has at most one extension to any open connected set in the plane, it becoems a challenge to find the alrgest open connected set into which it can be extended. this is called the theory of analytic continuation.
6) it turns out that one can continue a function along an arc by path integration along that arc. however if there are sevearl arcs leading from a starting point to the same end point, the integral along those twqo acrs may not yield the samke value, so analytic continuyation depends on the arc along which we choose to extend the function.
7) however if two arcs are smoothl;y deformable nito one another, the integral along both yields the same answer, this is called cauchy's integral theorem. thus one is led to construct an abstarct spave lying over the plane on which the naturalk alrgest extension of the opriginal function is =defined and analytic, called the riemann surface of the function. this space has one point for each deformation clas of arcs l;eading from the opriginal point to a given extension point.
8) at each "singularity", i.e. a point to which an analytic function cannot be extended, but such that the function can be extended to a punctured nbhd of this point, one can expand the function as a laurent series, i.e. a powers eries with also negative powers of z.
then there are two different behaviors, either there are infinitely many negative terms, otr only finitely many, if only finitely many then the function approaches infinity as z approaches the popint, thus the function extends continmuosly as a map to the sphere artehr than the plane.
if there are infinitely many, then the function has no limit as z approaches the point, and in fact on every nbhd of the point the function actually assumes all but at most two values.
9) for functions with only a finite number of negative terms to their power series (called meromorphic), the question arises which collections of negative terms, or "principal parts" can occur for some function, the so called mittag leffler problem. in the plane any collection of negative terms is possible for some function, but on compact riemann surfaces such as a torus (doughnut) or surface of higher genus, there is an obstruction. i.e. each meromorphic differential has at each singularity a "residue", the valoue of the coefficient of z^(-1), which is infact independent of the coordiantes used to describe the differential.
then on a compact riemann surface the sum of the residues must equal zero. indeed thius is the only obstruction to existence of a differential, and a slight generalization of this result is called the famous riemann roch theorem.
10) residues can also be used to compute complex and also real integrals. i.e. cauchy's theorem says the integral of a differential form around a loop with no singularities inside is zero, and the residue theoprem says that if there are residus then the integral equals (2pi i times) the sum of the residues.
11) we have seen that riemann surfaces arise from trying to extend locally defined analytic functions as far as possible. the resulting surface is compact in case we begin with a function defiend implicitly by a polynomial in two variables. riemann proved that the converse is also true, i.e. any compact rioemann surface has an analytic immersion in the projective plane.
this covers much mroe than the first 7 chapters of churchill.
as to conformla mapping, a conformal map is just a map defiend by analytic functions,a nd which is one to one and onto. riemanns big theorem is that any open set in the plane which is "simply connected": i.e. every loop in it can be shrunk to a point without passing outside the set, is actually conformally isomorphic to the open disc, or to th whole plane, (and these 2 cases are distinct).
it is sometimes of practical use in engineering to give an explicit conformal isomorphism between different looking sets, like the right upper quarter plane and the upper half plane (using an exponential function to change angles at the origin.)
