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Graduate Texts in Mathematics

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In the second part of the text, we continue with the leisurely exploration of interesting consequences and applications of the Fundamental Theorem. We are grateful to Lipman Bers for introducing us to the beauty of the theme.

The Fundamental Theorem in Complex Function Theory

Some motivation

Is it possible to divide the positive integers Z>0 into finitely many (more than 1) infinite arithmetic progressions with clear differences. 1.1) Choose a set of complex numbers2 {zk}with an absolute value less than 1 with lim. all these quantities are finite), while lim.

The Fundamental Theorem

It is, of course, possible to follow other paths through the various assertions to arrive at our main result. For the convenience of the reader, we describe where the various implications are.

Outline of text

Foundations

Introduction and preliminaries

The complex plane can be viewed as a subset of the complex sphere C that is C compressed by a neighboring point known as a point at infinity, such that C = C∪. A non-negative real number r is called the absolute value or norm or modulus of the complex number z.

Figure 2.1. The complex plane
Figure 2.1. The complex plane

Differentiability and holomorphic mappings

The function f is continuous and its real and imaginary parts satisfy the Cauchy-Riemann equations at z = 0, but it is not differentiable at z = 0. If the function f has continuous first partial derivatives in a neighborhood of c it satisfies the CR equations at c, then f is (complex) differentiable at c.

Power Series

Complex power series

As a result of the following theorem, it makes sense to define {z ∈C; |z|< ρ}as a power series convergence disk. Then the root test will hold with L = 14, but the sequence of ratios will obviously not converge.

More on power series

Looking at the diagrams in the (i, k) plane shown in Figure 3.1, we see that. n=0anzn has radius of convergence ρ >0. note that this defines a continuous function by Corollary 3.8). The results obtained so far have provided information about the behavior of a power series within its convergence disk.

The exponential function, the logarithm function, and some complex trigonometric functions

A continuous function f on a domain D that does not contain the origin is called a branch of the logarithm on D if we have for all z ∈D, ef(z) =z. Later we will establish that under appropriate conditions on D there always exists a branch of the logarithm.

An identity principle

A function f defined in a quarter of ζ∈C has a power series expansion at ζ if there exists an r >0 such that. Let f be a function defined in a quarter of ζ ∈C that has a power series expansion atζ with radius of convergence ρ. a) f is holomorphic and C∞ near ζ.

Figure 3.2. Radii of convergence
Figure 3.2. Radii of convergence

Zeros and poles

In cases (II) and (III) of the definition, h has a power series expansion at ζ due to the following. Find the radius of convergence of the power series ∞. a) Show that tanz = tanζ if and only if there exists an integer k such thatζ−z =πk.

The Cauchy Theory–A Fundamental Theorem

Line integrals and differential forms

If γ is a differentiable path in D and ω is a differentiable form in D, then we define the orpath line or contour integral of ω along γ, by the formula. We say that γ is a piecewise differentiable path (henceforth abbreviated pdp) if there exists a partition of [a, b] of the form given in (4.1) such that every path defined by (4.2) is differentiable. The path integral is well defined (independent of separation) and agrees with the previous definition for differentiable paths.

Figure 4.1. Three figures
Figure 4.1. Three figures

The precise difference between closed and exact forms Although the definitions of exact and closed are straightforward,

The first of the two real forms is exact, and the second is closed but not exact in D. Note also that argz and arctanxy are multivalued functions whose differences agree and are single-valued. If γ is a continuous path in a domain D and ω is a closed form in D, then there exists a primitive f of ω along γ;f is unique up to the addition of a constant.

Figure 4.2. The integration path for F
Figure 4.2. The integration path for F

Integration of closed forms and the winding number Consideration of the next example leads to the extension of the

A region D ⊆ C is simply said to be connected if every closed path in D is homotopic to a point in D. 1) The complex plane C is simply connected. In a simply connected domain, a differential form is closed if and only if it is exact. In any simply connected domain that does not contain the point 0, there exists a branch of logz.

Figure 4.3. Homotopy with fixed end points 4.4. Homotopy and simple connectivity
Figure 4.3. Homotopy with fixed end points 4.4. Homotopy and simple connectivity

Winding number

Note that the above definition of a product of two paths differs from that used in topology, where paths are traversed in succession, but at twice the speed.4.

Cauchy Theory: initial version

The second equality follows from the fact that certain paths on the boundaries of the subrectangles have opposite directions, giving cancellations in the integral (see Figure 4.5). If f is holomorphic to an open group and γ is a continuous closed path in D that is homotopic to a point in D, then. C(z −z0)mdz directly from the definition, where C is the circle centered at z0 with radius r > 0 and.

Figure 4.5. The integrals along the common side of R 1 and R 2 are in opposite directions
Figure 4.5. The integrals along the common side of R 1 and R 2 are in opposite directions

The Cauchy Theory–Key Consequences

  • Consequences of the Cauchy Theory
  • Cycles and homology
  • Jordan curves We introduce some more terminology
  • The Mean Value Property
  • On elegance and conciseness
  • Appendix: Cauchy’s integral formula for smooth functions

The function must of course be holomorphic in a quarter of the closed disc bounded by the circle. We will consider the interval of γ as the union of the intervals of the components γi. From the above, it is easy to deduce that a non-constant holomorphic function on a bounded domain (since it satisfies MVP) that is continuous at the closure of the domain assumes its maximum on the boundary of that domain.

Figure 5.1. The rectangles R and R δ
Figure 5.1. The rectangles R and R δ

Cauchy Theory: Local Behavior and Singularities of Holomorphic Functions

Functions holomorphic on an annulus

Isolated singularities

If g were restricted to such a neighborhood, it would have a mobile singularity at z= 0 and thus extend to a holomorphic function on U(0, ε); therefore f will. Place a small positively oriented circle around each zj and use the extended version of Cauchy's theorem.

Zeros and poles of meromorphic functions

To explain the name of the theorem, we note that the principle of the argument can be expressed as follows: Let D be a domain in C and let f ∈ M(D). If Z denotes the number of zeros of the function f − c inside γ (multiplicity count) and P denotes the number of poles of f inside γ (multiplicity count), then the argument f − c increases by 2π(Z−P) when crossing γ. The first integral on the far right side of the equation is zero because it is log|f−c| has one value.

Local properties of holomorphic maps

Much of the above discussion, as well as the next corollary, are slight amplifications of the material in the previous section. All uses of the multivalued arg function must be interpreted correctly; we leave that to the reader to do. The statement about lengths means that the ratio of the length of Δw to the length of Δz tends to |f(z0)| as z tends to toz0.

Evaluation of definite integrals

Q(x) dx, where Q is a rational function with no singularities on R and with ν∞(Q)≥2. 2) A second class of integrals that can be evaluated by the Residue Theorem consists of those of the form Next, we estimate the integral over γ5:. because g is bounded on a neighborhood of 0 and the length of the path of integration goes to zero. Suppose f is meromorphic in a neighborhood of the closed unit disc, that it has no zeros or poles in the unit disc, and that |f(z)|= 1 for |z| = 1.

Figure 6.1. The path of integration for Example (1)
Figure 6.1. The path of integration for Example (1)

Sequences and Series of Holomorphic Functions

Consequences of uniform convergence on compact sets We begin by recalling some notation and introducing some new

If {fn} ⊂ H(D) and {fn} converge uniformly on all compact subsets of D, then the limit function f is holomorphic on D. If {fn} ⊂ H(D) and fn → f uniformly on all compact subsets of D, then fn → f uniform on all compact subsets of D. Let {fn} be a series of holomorphic functions on D such thatfn →f uniform on all compact subsets of D.

A metric on C(D)

If {fn} is a sequence in H(D) with fn → f uniformly on all compact subsets of D and fn is free for every n, then f is constant or free. Convergence in the ρ-metric in C(D) is equivalent to uniform convergence on all compact subsets of D. Indeed, we have shown more than we claim: if {fn} is a ρ-Cauchy sequence in C(D), then there exists f ∈ C(D ), so that fn → f is uniform on all compact subsets of D.

The cotangent function

The last sum is clearly zero, and the residual of G at 0 is 0 because G is an even function. So there exists an M > 0 such that |cotπt| ≤ M for t on the horizontal sides and the claim is proven.

Figure 7.1. The square C N
Figure 7.1. The square C N

Compact sets in H(D)

Then the sequence {fk} converges uniformly on all compact subsets of D if and only if lim. Iffk→f uniformly on all compact subsets of D, then for every nonnegative integer n,fk(n)→f(n) uniformly on compact subsets of D; in particular fk(n)(ζ)→f(n)(ζ), since a set consisting of one point is definitely compact. 7.4) Assertion (7.4) is sufficient to prove the statement; because then by Lemma 7.27 the sequence {fk}k∈B converges uniformly on the closed disc clU(zi, ri) for every i, which implies that the same sequence converges uniformly on all compact subsets of D.

Approximation theorems and Runge’s theorem In this part of the chapter we consider the problem of approximating

We begin with some preliminary investigations from real analysis needed in the proof of Runge's theorem. We now prove that these γi also satisfy equation (7.5), where the equation involves the function f given in the statement of the lemma. If is any simply connected domain in the plane and f is a holomorphic function in D, then f can be approximated uniformly on all compact subsets of D by polynomials.

Conformal Equivalence

Fractional linear (M¨ obius) transformations

  • cross-ratios

Also, when convenient, we will multiply each of the four constants a, b, c, and d by −1, since this does not change the M¨obius transformation nor the condition ad−bc= 1. That is, for each pair of consecutive cards in the suit, the core of the second card coincides with the image of the first. It is also clear that the image of the last arrow in the series (8.2) is exactly the M¨obius group and is therefore isomorphic to PSL(2,C), the quotient of SL(2,C) by ±I as defined above.

Figure 8.1. The cross-ratio arguments
Figure 8.1. The cross-ratio arguments

The Riemann Mapping Theorem

Just as we defined PSL(2,C) in section 8.1 and then proved that it is isomorphic to the group Aut(C), we can use the group PSL(2,R) = SL(2,R)/{± I define } of appropriate matrices with real coefficients modulo plus or minus the identity matrix and obtain the following description. The mapg1 is the map f followed by a rotation through the angle −θ and sends w0 tor. If D is a simply connected domain in C, then D is conformally equivalent to one and only one of the following: (i) C, (ii) C, or (iii) D.

Hyperbolic geometry

  • The Poincar´ e metric

For any two points z and w in H2, the hyperbolic length of the geodesic segment γ joining z and w is given by. Holomorphic self-maps of the unit disk do not increase the distance in the hyperbolic metric. A holomorphic self-map of the unit disk is either an isometry or a contraction with respect to the hyperbolic metric.

Figure 8.2. In the upper half-plane
Figure 8.2. In the upper half-plane

Finite Blaschke products

Then, as a result of the last statement, and by substituting f◦T and B ◦T, replacing f and B with f◦T and B ◦T respectively, we can assume that a0 = 0. Formulate and prove (as a result of the Riemann Mapping Theorem 8.20) the Riemann Mapping Theorem for simply connected domains D⊆C. Show that {fn} contains a subsequence that uniformly converges on compact subsets of the unit disk.

Harmonic Functions

  • Harmonic functions and the Laplacian We begin with
  • Integral representation of harmonic functions We begin to apply the Cauchy Theory toward our present main
  • The Dirichlet problem
    • Fourier series interpretation of the Poisson formula
  • The Mean Value Property: a characterization Harmonic functions satisfy the MVP, as we have seen in Corollary
  • The reflection principle

Furthermore, if D is bounded and f is continuous on the closure of D, with m ≤f ≤ M on ∂D for some real constants m and M, then m≤f ≤M onD. In this case, P[u] is obviously the real part of an analytic function on disk. Thus, we have the following procedure for expanding a given continuous function u on the unit circle into a continuous function u.

Figure 9.1. u 1 and u 2
Figure 9.1. u 1 and u 2

Zeros of Holomorphic Functions

Infinite products

數據

Figure 2.1. The complex plane
Figure 3.1. The (i, k) plane
Figure 3.2. Radii of convergence
Figure 4.1. Three figures
+7

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