Review: Complex Analysis In the Spirit of Lipman Bers | EMS
Remark 5. Recall that we have established the estimates 5. An immediate corollary is Corollary 5. A bounded entire function is constant. Use the Taylor series expansion of the function at the origin and the estimate 5. Definition 5. The paths in a cycle are not necessarily distinct. We extend the notion of the integral of a function over a single closed path to the integral over a cycle as follows: Definition 5.
A topologist would develop the concept of homology in much more detail using chains and cycles. In particular, our cycles are allowed repetitions of the component curves, and the component curves may be taken in any order since we are only concerned with the sum of the integrals.
Jordan curves We introduce some more terminology. Since the circle is compact, h is a homeomorphism. We state, without proof, Theorem 5. We shall not prove the above theorem. It is a deep result. In all of our applications, it will be obvious that our Jordan curves have the above properties. This observation allows us to prove Theorem 5. IX of J. X, Academic Press, , or Chapter 8 of J. Munkres, Topology, a first course, Prentice-Hall Inc. We can apply Theorem 5. The Mean Value Property The next concept applies in a broader context than that of holomorphic functions, as we will see in Chapter 9.
If f is a nonconstant holomorphic function on a domain D, then f z has no maximum in D. Furthermore, if D is bounded and f is continuous on the boundary of D, then f z assumes its maximum on the boundary of D. By studying the proof of Theorem 5. An important consequence of Corollary 5. Now for any r with 0 96 5. On elegance and conciseness The Jordan Curve Theorem is a major result in two-dimensional topology. All the other theorems and corollaries of this chapter are milestones in function theory.
Folland Advanced Calculus, Prentice Hall, , one establishes we are using complex notation the following form of Theorem 5. Exercises 5. Prove that the following conditions are equivalent. How unique are the functions g and h? Is the converse true? Let f be a holomorphic function on z for all z 98 5. Prove that f is a polynomial of degree at most b. Show that f is identically 0. Let D be a bounded domain in C. Prove the Maximum and the Minimum Principles stated in Remark 5. Functions that are holomorphic on an annulus have Laurent series expansions, an analog of power series expansions for holomorphic functions on disks.
We discuss the local properties of these functions. The theorem is not only aesthetically pleasing in its own right but also allows us to give alternative proofs of many important results. Functions holomorphic on an annulus Theorem 6. It is uniquely determined by f and A.
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Corollary 6. Existence: Already done.
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Example 6. Theorem 6. This is usually done in more advanced text books. Definition 6. Let f be holomorphic in a domain D except for isolated singularities at z1 ,. The theorem then follows immediately from 6. Thus has residue n at 0.
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An injective holomorphic function is a homeomorphism from its domain onto its image. See also the discussion in the next section. Let f and g be holomorphic functions on a domain D.
Recall from Theorem 6. In this section we describe the behavior of f near z0 using results from the previous section. Proposition 6.
Let D be a domain in C, z0 a point in D and f a function holomorphic on D. The existence of h is a consequence of Exercise 5. By 1 , g1 is a homeomorphism from a neighborhood of z0 to a neighborhood of 0. Remark 6. The above property 2 of holomorphic mappings is also a consequence of Corollary 6. An immediate consequence of 1 and 2 is the following Corollary 6.
All uses of the multivalued arg function need to be interpreted appropriately; we leave it to the reader to do so. We illustrate this with a few examples. We illustrate with a more complicated example, where Q has a simple pole at the origin.
Here the ordinary integral is replaced by its principal value pr. Which are the possible values for f 0 and for f 17? Suppose f is holomorphic for z 7 n3 for 6. Compute f 8 Evaluate z tan z dz. If f is holomorphic on 0 0 , what kind of singularity does f have at 0?
Let D be an open, bounded, and connected subset of C with smooth boundary. Show that f is constant. Prove the following extension of the Maximum Modulus Principle. Is there a 6. We then apply these ideas to obtain a series expansion for the cotangent function. In the fourth section, we characterize the compact subsets of the space of holomorphic functions on a domain. This powerful characterization is used in Section 7.
This characterization will also be used in Chapter 8 to prove the Riemann Mapping Theorem. Consequences of uniform convergence on compact sets We begin by recalling some notation and introducing some new symbols. Proposition 7. The converse is not true: Uniform convergence on all compact subsets of D is stronger than pointwise convergence.here
To see this observe that we know from Theorem 2. Consider any sequence of continuous functions converging at every point of the domain to a discontinuous function. Such an example is easily constructed see Exercise 7. We now study some consequences of this notion of uniform convergence on compact subsets of D, also called locally uniform convergence, for H D. Corollary 7. The next consequence of uniform convergence on compact sets is that uniform convergence of a sequence of holomorphic functions on compact subsets implies uniform convergence of the derivatives on compact subsets.
We leave this construction to the reader as Exercise 7. We call f simple, univalent, or schlicht if it is one-to-one injective on D. Remark 7.