Necessary Complex analysis...

Let U be an open set in C^{n} and f: U--> C be a C^{1}-map. Let u be in U and let T_{U,u} denote the tangent space to U at u. (Note: there is a canonical isomorphism between T_{U,u} and C^{n}.)

Def: A function f is holomorphic if for all u in U the differential df_{u} in Hom(T_{U,u},C)=Hom(C^{n},C) is C-linear. Equivalently f is killed by differentiation w.r.t the complex conjugate to each coordinate z_{i} of C^{n}.

We note that the set of all holomorphic functions forms a ring: if f is a holomorphic function that does not vanish on an open set U then 1/f is holomorphic and if f and g are both holomorphic functions then equally f+g and fg are.

It is also worth noting that composition of holomorphic functions also yields a holomorphic function.

More often than not the term "holomorphic" and "complex-analytic" are used interchangably and this is a result of the, not exactly trivial, theorem:

Theorem: Holomorphic functions in complex varaible z1, z2 ,.., zn admit expansions as power series in variables zi.

There are two particularly important complex analytic results which are studied very early on in an undergraduate course: Stoke's Theorem and Cauchy's theorem. Stoke's theorem (as I hope to get on to at some point) is important as it is used in Algebraic Geometry to pair de Rham cohomology and singular homology and Cauchy's theorem is important as it is used to obtain some very useful analytic continuation results.

Theorem: Let U be an open connected set of C^{n} and let f be a holomorphic function on U. If f vanishes on an open set of U then f is identically zero.

Riemann Extension Theorem: Let f be a bounded holomorphic function defined on the complement of a set {z : z1=0} in U, where U is an open in C^{n}. Then f extends to a holomorphic map defined on U.

Hartog's Extension Theorem: Let U be an open set of C^{n} and f a holomorphic function on the complement of a set S={z : z1=z2=0} in U. Then f extends to a holomorphic map defined on U.

We note that Hartog's Extension theorem also holds if S is of codimension 2 in U.

So what can we get from these extension theorems? They tell us that possible singularities of a holomorphic function can not exist unless the function is not bounded and are not defined on the complement of some analytic subset of codimension 2.