复分析代考_Complex function代考_MATH307

复分析代考_Complex function代考_The Cauchy Integral Formula for Multiply Connected Domains

In previous chapters we treated the Cauchy integral formula for holomorphic functions defined on holomorphically simply connected (h.s.c.) domains. (In Chapter 4 we discussed the Cauchy formula on an annulus, in order to aid our study of Laurent expansions.] In many contexts we focused our attention on discs and squares. However it is often useful to have an extension of the Cauchy integral theory to multiply connected domains. While this matter is straightforward, it is not trivial. For instance, the holes in the domain may harbor poles or essential singularities of the function being integrated. Thus we need to treat this matter in detail.

The first stage of this process is to notice that the concept of integrating a holomorphic function along a piecewise $C^1$ curve can be extended to apply to continuous curves. The idea of how to do this is simple: We want to define $\oint_\gamma f$, where $\gamma:[a, b] \rightarrow \mathbb{C}$ is a continuous curve and $f$ is holomorphic on a neighborhood of $\gamma([a, b])$. If $a=a_1<a_2<\cdots<a_{k+1}=b$ is a subdivision of $[a, b]$ and if we set $\gamma_i$ to be $\gamma$ restricted to $\left[a_i, a_{i+1}\right]$, then certainly we would require that
$$\oint_\gamma f=\sum_{i=1}^k \oint_{\gamma_i} f$$
Suppose further that, for each $i=1, \ldots, k$, the image of $\gamma_i$ is contained in some open $\operatorname{disc} D_i$ on which $f$ is defined and holomorphic. Then we would certainly want to set
$$\oint_{\gamma_i} f=F_i\left(\gamma_i\left(a_{i+1}\right)\right)-F_i\left(\gamma_i\left(a_i\right)\right)$$
where $F_i: D_i \rightarrow \mathbb{C}$ is a holomorphic antiderivative for $f$ on $D_i$. Thus $\oint_\gamma f$ would be determined. It is a straightforward if somewhat lengthy matter (Exercise 36) to show that subdivisions of this sort always exist and that the resulting definition of $\oint_\gamma f$ is independent of the particular such subdivision used. This definition of integration along continuous curves enables us to use, for instance, the concept of index for continuous closed curves in the discussion that follows. [We take continuity to be part of the definition of the word “curve” from now on, so that “closed curve” means continuous closed curve, and so forth.] This extension to continuous, closed curves is important: A homotopy can be thought of as a continuous family of curves, but these need not be in general piecewise $C^1$ curves, and it would be awkward to restrict the homotopy concept to families of piecewise $C^1$ curves.

复分析代考_Complex function代考_Holomorphic Simple Connectivity

In Section $6.7$ we proved that if a proper subset $U$ of the complex plane is holomorphically (analytically) simply connected, then it is conformally equivalent to the unit disc. At that time we promised to relate holomorphic simple connectivity to some more geometric notion. This is the purpose of the present section.

It is plain that if a domain is holomorphically simply connected, then it is (topologically) simply connected. For the hypothesis implies (by the analytic form of the Riemann mapping theorem, Theorem 6.6.3) that the domain is either conformally equivalent to the disc or to $\mathbb{C}$. Hence it is certainly topologically equivalent to the disc. Therefore it is simply connected in the topological sense.

For the converse direction, we notice that if a domain $U$ is (topologically) simply connected, then simple modifications of ideas that we have already studied show that any holomorphic function on $U$ has an antiderivative: Suppose that $f: U \rightarrow \mathbb{C}$ is holomorphic. Choose a point $P \in U$. If $\gamma_1, \gamma_2$ are two piecewise $C^1$ curves from $P$ to another point $Q$ in $U$, then $\oint_{\gamma_1} f(z) d z=\oint_{\gamma_2} f(z) d z$. This equality follows from applying Corollary 11.2.5 to the (closed) curve made up of $\gamma_1$ followed by ” $\gamma_2$ backwards” (this curve is homotopic to a constant, since $U$ is simply connected). Define $F(Q)=\oint_\gamma f(z) d z$ for any curve $\gamma$ from $P$ to $Q$. It follows, as in the proof of Morera’s theorem (Theorem 3.1.4), that $F$ is holomorphic and $F^{\prime}=f$ on $U$.

Thus one has the chain of implications for a domain $U \subset \mathbb{C}$ but unequal to $\mathbb{C}$ : homeomorphism to the $\operatorname{disc} D \Rightarrow$ (topological) simple connectivity $\Rightarrow$ holomorphic simple connectivity $\Rightarrow$ conformal equivalence to the disc $D$ $\Rightarrow$ homeomorphism to the disc $D$, so that all these properties are logically equivalent to each other. [Note: It is surprisingly hard to prove that “simple connectivity” $\Rightarrow$ “homeomorphism to $D$ ” without using complex analytic methods.] This in particular proves Theorem 6.4.2 and also establishes the following assertion, which is the result usually called the Riemann mapping theorem: Theorem 11.3.1 (Riemann mapping theorem). If $U$ is a connected, simply connected open subset of $\mathbb{C}$, then either $U=\mathbb{C}$ or $U$ is conformally equivalent to the unit disc $D$.

复分析代考

复分析代考_Complex function代考_The Cauchy Integral Formula for Multiply Connected Domains

$$\oint_\gamma f=\sum_{i=1}^k \oint_{\gamma_i} f$$

$$\oint_{\gamma_i} f=F_i\left(\gamma_i\left(a_{i+1}\right)\right)-F_i\left(\gamma_i\left(a_i\right)\right)$$

复分析代考_Complex function代考_Holomorphic Simple Connectivity

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