We continue with a discussion about meromorphic functions and the properties of analytic functions. Later notes will consider the Riemann mapping theorem, harmonic functions and the Dirichlet problem among other topics.
Definition 2.1 A function on an open set is meromorphic if there exists a discrete set of points such that is holomorphic on and has poles at each . Furthermore, is meromorphic in the extended complex plane if is either meromorphic or holomorphic at . In this case we say that has a pole or is holomorphic at infinity.
By collecting results from the previous section, we are immediately led to the following proposition regarding the Laurent expansions of complex valued functions.
Proposition 2.2 Let be the discrete set of singularities of a complex function where is an open set in . For a fixed , suppose the Laurent expansion for in an annulus about is given by . Then,
- The function has a removable singularity at if and only if for all .
- The function has a pole at if and only if there exists with such that for all ; that is, the Laurent expansion of about has only finitely many negative terms.
- The function has an essential singularity at if and only if the Laurent expansion of about has infinitely many negative terms.
- Furthermore, is meromorphic on the extended complex plane if and only if there exists such that for .
Proof: The first three results follow immediately from the previous section. For the final result, consider the function . Suppose has an essential singularity at infinity. Then, has an essential singularity at the origin and item 3 and must have infinitely many negative terms. Substituting for , it follows that must have infinitely many positive terms. The other direction of this statement is similar, and our conclusion follows. .
These results lead to an impressive description of meromorphic in the extended complex plane.
Theorem 2.3 Suppose is a meromorphic function in the extended complex plane. Then is a rational function.
Proof: Let be the set of poles of the function . The function must be analytic in a deleted neighborhood of the origin, hence is analytic in a deleted neighborhood of ; the remaining region of the complex plane can contain only finitely many singularities, therefore must be finite. Let denote the principal part of the Laurent expansion of at the pole . In addition, let denote the principal part of the Laurent expansion of at the point at infinity. Our study of the singularities of complex functions and the Laurent expansion demonstrates that for all , the principal parts are polynomials of finite length in ; the same holds for . Furthermore, we have
is entire and bounded, hence constant. It follows that is a rational function, as desired.
A critical result in the theory, enabled in part by the Laurent expansion allows one to calculate complex line integrals with relative ease. First, we note that there is little in the development of Cauchy’s integral formula that seems to imply that a holomorphic function is defined by its integral about a circle. In addition, Goursat’s theorem can easily be extended to rectangles and more complex polygons as well. These results seem to imply that we can extend Cauchy’s theorem and therefore much of our development of the theory to arbitrary rectifiable closed curves. Doing so will enable us to study more challenging questions with a greater number of tools at our disposal; for example, the desired extension will allow us to analytically continue Riemann’s zeta function to the entire complex plane from the small set on which it is originally holomorphic. We begin with a description of the region enclosed by a rectifiable closed curve.
Definition 2.4 Given a rectifiable closed curve and a point the winding number of about is given by the formula
Theorem 2.5 Under the assumptions above, the following statements regarding the winding number hold:
- The map is constant inside the connected component of containing .
- as . In other words, is the unbounded, connected component of .
Before we proceed with a proof of the theorem, we note that the definition of the winding number resembles the complex logarithm. Indeed, the properties of the complex logarithm motivate the definition. Without excising a branch of the complex plane, the complex logarithm is not a well defined function. Suppose and write for some and . Then,
We can produce a holomorphic, well defined complex logarithm on by defining a branch cut ; that is, by removing a set (generally an appropriate path) from and restricting to . However, without doing so, the value of the logarithm differs by an integral factor of . If we begin at the point , traverse the unit circle, and return to , the complex logarithm at will differ by . Traversing the unit circle again, we find that the complex logarithm now differs by . It is with this in mind that we have proposed the above definition and prove the theorem. Proof:
- To simplify our argument, suppose is smooth. If it not, we subdivide into intervals disjoint invervals such that is left and right differntiable at . Our argument then proceeds with slight modification. Consider and let
Then, is continuous on and differentiable with . We hope to show that for some . To this end, we write
Therefore, such that . In other words, . Since is closed, , hence it follows that
Finally, we have which implies ; that concludes the proof of our theorem.
- We first show that is continuous inside the connection component of . This, paired with the fact that is integer valued on the connected component proves that it must be constant there. Consider both inside the connected component of . We have,
Since stays away from both , find that the denominator of the integrand is bounded, hence
where and is the length of the curve . Our assertion follows immediately.
- Suppose is an element of the unbounded connected component of and that is large. Then
must tend to zero since becomes arbitrarily large. Again, the assertion is immediate.
We are now able to prove a general version of the celebrated residue theorem. Our derivation of the residue formula and proof of the theorem will be based on the equivalence of a collection of four statements. After exhibiting these equivalences, we will prove the general Cauchy integral theorem for simply connected sets, which will imply the first of these statements. As a result, the residue theorem will hold on simply connected sets, and we will use it to state and prove a variety of properties of analytic functions.
Definition 2.6 Suppose is a meromorphic function on the set with poles at . Let
The coefficient is called the residue of at and is denoted
Theorem 2.7 Suppose has a simple pole, (i.e. a pole of order ) at . Then
Instead, if has a pole of order at , then,
Proof: Both statements follow directly from the Laurent expansion of at .
Theorem 2.8 The residue theorem Suppose is a rectifiable, closed curve in an open set , is holomorphic on and is meromorphic on . Then, the following statements are equivalent:
Proof: First, we prove that statement implies statement Let
Then, we find that
Since is holomorphic and its derivative is bounded, must be holomorphic in , hence by statement , .
On the other hand, let . Clearly is holomorphic, and statement then implies
As , it follows that statement implies .
Next, suppose . Then the function is holomorphic on the set . By statement , we find that
hence ; that is, statement implies statement .
Now we prove that statement implies statement . Often this equivalence itself is known as the Residue theorem.