Line integrals: 3. Applications to complex analysis

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\newcommand{\bbc}{\mathbb{C}}
\newcommand{\Int}{\operatorname{Int}}
\)Navigation: 1. Exact, conservative and closed forms | 2. Locally exact forms and singular homology | 3. Applications to complex analysis

Preliminaries

Let \(F\) be a complex Banach space, let \(U\subseteq\bbc\) be an open set, and let \(f:U\to F\). Recall that the complex Fréchet derivative of \(f\) at \(z\in U\), if it exists, is a \(\bbc\)-linear map \(Df(z):\bbc\to F\). Not all real differentiable functions on \(\bbc\) are complex differentiable: for example, the (real) derivative of \(f(z)=\overline{z}\) (i.e. \(f(x+yi)=x-yi\)) at any point is represented by the matrix $$\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix},$$ but \(Df(z)\) is clearly not a \(\bbc\)-linear map.

Since \(\bbc\) is a one-dimensional vector space over itself, \(Df(z)\) is completely determined by the value \(Df(z)(1)\in F\): $$
Df(z)w=wDf(z)(1).
$$ We can therefore identify \(Df(z)\) with \(Df(z)(1)\), and from now on we will use the notation \(f'(z)=Df(z)(1)\in F\). For example, if \(f(z)=z^2\) then the Fréchet derivative is \(Df(z)w=2zw\), and \(f'(z)=Df(z)(1)=2z\) as usual.

Theorem 18. Let \(U\subseteq\bbc\) be an open set. A function \(f:U\to F\) is complex differentiable at \(z\in U\) if and only if the limit $$c=\lim_{h\to 0}\frac{f(z+h)-f(z)}{h}$$ exists. In that case, \(c=f'(z)\).

Proof. We have $$
\lim_{h\to 0}\frac{f(z+h)-f(z)}{h} = c \Leftrightarrow \lim_{h\to 0}\frac{f(z+h)-f(z)-ch}{|h|} = 0.
$$ \(\square\)

If \(f:U\to F\) is complex differentiable at all \(z\in U\), then we say that \(f\) is holomorphic on \(U\) or simply holomorphic. If \(A\subseteq\bbc\) is any set, we say that \(f\) is holomorphic on \(A\) if it is holomorphic on an open set containing \(A\). If \(z\in\bbc\) and \(f\) is holomorphic on a neighborhood of \(z\), we say that \(f\) is holomorphic at \(z\).

Using the basic properties of the derivative, we have:

Theorem 19 (Chain rule). Let \(U,V\subseteq\bbc\) be open sets. Let \(f:U\to\bbc\) and \(g:V\to F\) with \(f(U)\subseteq V\). If \(f\) is complex differentiable at \(z\) and \(g\) is complex differentiable at \(f(z)\), then \(g\circ f\) is complex differentiable at \(z\) and $$(g\circ f)'(z)=g'(f(z))f'(z).$$

Theorem 20. Let \(U\subseteq\bbc\) be an open set, let \(F_1,F_2\) be complex Banach spaces, and let \(f:U\to F_1\) and \(f:U\to F_2\) be complex differentiable at \(z\in U\).

  1. If \(f\) is constant then \(f'(z)=0\).
  2. If \(F_1=F_2\) then \((f+g)'(z)=f'(z)+g'(z)\).
  3. \((cf)'(z)=cf'(z)\) for all \(c\in\bbc\).
  4. If \(F_1=\bbc\) or \(F_2=\bbc\), then \((fg)'(z)=f'(z)g(z)+f(z)g'(z)\).
  5. If \(F_2=\bbc\) and \(g(z)\ne 0\) then \((f/g)'(z)=[f'(z)g(z)-f(z)g'(z)]/g(z)^2\).

Theorem 21. Let \(U\subseteq\bbc\) be an open set. A function \(f:U\to\bbc\) is complex differentiable at \(z\in U\) if and only if it is real differentiable at \(z\) and the real derivative \(Df(z)\) is represented by a matrix of the form $$\begin{bmatrix}a & -b \\ b & a\end{bmatrix}.$$ In that case, \(Df(z)\) is the matrix representation of the complex number \(f'(z)\). Furthermore, \(\det Df(z)=|f'(z)|^2\).

The preceding theorem shows that a holomorphic function \(f:U\to\bbc\) is simply a real differentiable function with the property that its derivative is a scalar times a rotation matrix at every point of \(U\). (That is, the derivative is \(\bbc\)-linear.) Suppose that \(f(x+yi)=u(x,y)+v(x,y)i\) where \(u,v\) are real valued functions. If \(f\) is holomorphic then the theorem implies that $$
\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\quad\mathrm{and}\quad\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}.
$$ These are known as the Cauchy-Riemann equations.

Integration

We now define complex line integrals as in part 1, taking \(E=\bbc\). If \(U\subseteq\bbc\) is an open set and \(f:U\to F\) is continuous, then we define its associated form \(\omega_f:U\to L(\bbc,F)\) by $$\omega_f(z)w=wf(z).$$ If \(\gamma\) is a curve in \(U\) then the integral of \(f\) along \(\gamma\) is defined by $$
\int_\gamma f = \int_\gamma f(z)\,dz = \int_\gamma \omega_f.
$$ Note that if \(\gamma:[a,b]\to U\) is a curve with partition \(\{a_0,\dots,a_k\}\), then $$
\int_\gamma f = \sum_{i=1}^k \int_{a_{i-1}}^{a_i} f(\gamma(t))\gamma'(t)\,dt.
$$ The usual properties in Theorem 1 apply. A holomorphic function \(g:U\to F\) satisfying \(f=g’\) is called a primitive of \(f\). It is easy to check that any potential function for \(\omega_f\) (a function \(g\) such that \(\omega_f=Dg\)) is a primitive for \(f\).

Example 22. Let \(n\) be an integer and define a curve \(\gamma:[0,2\pi]\to\bbc\) by \(\gamma(t)=e^{it}\). Then \begin{align}
\int_\gamma z^n\,dz &= i\int_0^{2\pi} e^{(n+1)it}\,dt \\
&= \begin{cases}2\pi i, & n=-1, \\ 0, & n\ne -1.\end{cases}
\end{align}

Suppose \(f:U\to F\) is holomorphic. Since \(\omega_f=\ell\circ f\) where \(\ell:F\to L(\bbc,F)\) is the linear map given by \(\ell(x)h=hx\), we have \begin{align}
D\omega_f(z)(u,v) &= (D\ell(f(z))Df(z)u)(v) \\
&= \ell(Df(z)u)(v) \\
&= uvf'(z).
\end{align} Clearly, \(D\omega_f(z)\) is symmetric for all \(z\in U\). Thus \(\omega_f\) is closed, and Goursat’s theorem (Corollary 8) shows that \(\omega_f\) is locally exact. As a consequence, we can define the integral of a holomorphic function \(f\) along any path (see Lemma 9).

Theorem 23 (Cauchy’s theorem, local version). Let \(U\subseteq\bbc\) be an open set, let \(\gamma_1,\gamma_2\) be paths in \(U\) that are homotopic, and let \(f\) be holomorphic on \(U\). Then $$\int_{\gamma_1} f = \int_{\gamma_2} f.$$ In particular, if \(U\) is simply connected then $$\int_\gamma f=0$$ for any closed path \(\gamma\) in \(U\).

Proof. Apply Theorem 10. \(\square\)

If \(C\) is a circle, we write $$\int_C f$$ for the integral of \(f\) along \(C\), taken counterclockwise.

Theorem 24 (Cauchy’s integral formula, local version). Let \(D\) be a closed disc and let \(f\) be holomorphic on \(D\). Then $$f(z)=\frac{1}{2\pi i}\int_{\partial D} \frac{f(\zeta)}{\zeta-z}\,d\zeta$$ for every \(z\in\Int D\).

Proof. Let \(U\) be an open set containing \(D\) on which \(f\) is holomorphic. For small \(r > 0\), the circle \(C_r\) of radius \(r\) around \(z\) is contained in \(D\). Note that \(C_r\) is homotopic to \(\partial D\) in \(U\setminus\{z\}\), so using Example 22 and Theorem 23 we have \begin{align}
\left\vert \frac{1}{2\pi i}\int_{\partial D}\frac{f(\zeta)}{\zeta-z}\,d\zeta-f(z)\right\vert &= \left\vert \frac{1}{2\pi i}\int_{C_r}\frac{f(\zeta)}{\zeta-z}\,d\zeta – \frac{1}{2\pi i}\int_{C_r}\frac{f(z)}{\zeta-z}\,d\zeta \right\vert \\
&= \left\vert \frac{1}{2\pi i}\int_{C_r}\frac{f(\zeta)-f(z)}{\zeta-z}\,d\zeta\right\vert \\
&\le \frac{1}{2\pi} 2\pi r \sup_{\zeta\in C_r} \left\vert \frac{f(\zeta)-f(z)}{\zeta-z} \right\vert \\
&\to 0
\end{align} as \(r\to 0\) since \(f\) is complex differentiable at \(z\). \(\square\)

The open disc \(D_r(z_0)\) is the set \(\{z\in\bbc:|z-z_0| < r\}\), and the closed disc \(\overline{D}_r(z_0)\) is the set \(\{z\in\bbc:|z-z_0| \le r\}\).

Theorem 25. Let \(D=\overline{D}_r(z_0)\) be a closed disc and let \(f\) be holomorphic on \(D\). Then $$
f(z)=\sum_{n=0}^\infty a_n(z-z_0)^n \tag{*}
$$ for every \(z\in\Int D\), where $$
a_n = \frac{1}{2\pi i} \int_{\partial D} \frac{f(\zeta)}{(\zeta-z_0)^{n+1}}\,d\zeta = \frac{1}{n!} f^{(n)}(z_0).
$$ We have $$
|a_n| \le \frac{1}{r^n} \sup_{\zeta\in\partial D} |f(\zeta)|,
$$ so the power series in (*) has a radius of convergence of at least \(r\).

Proof. By Theorem 24, we have $$f(z)=\frac{1}{2\pi i}\int_{\partial D} \frac{f(\zeta)}{\zeta-z}\,d\zeta.$$ Let \(0 < s < r\) and let \(D'=\overline{D}_s(z_0)\). For all \(z\in D'\) and \(\zeta\in\partial D\) we have \begin{align} \frac{1}{\zeta-z} &= \frac{1}{\zeta-z_0}\left(\frac{1}{1-\frac{z-z_0}{\zeta-z_0}}\right) \\ &= \frac{1}{\zeta-z_0}\sum_{n=0}^\infty \left(\frac{z-z_0}{\zeta-z_0}\right)^n, \end{align} where the geometric series converges absolutely and uniformly for \(\zeta\in\partial D\) since $$ \left\vert\frac{z-z_0}{\zeta-z_0}\right\vert \le \frac{s}{r} < 1. $$ Therefore \begin{align} f(z) &= \frac{1}{2\pi i}\int_{\partial D} \frac{f(\zeta)}{\zeta-z_0} \sum_{n=0}^\infty \left(\frac{z-z_0}{\zeta-z_0}\right)^n\,d\zeta \\ &= \sum_{n=0}^\infty \left[ \frac{1}{2\pi i}\int_{\partial D}\frac{f(\zeta)}{(\zeta-z_0)^{n+1}}\,d\zeta \right] (z-z_0)^n \end{align} for all \(z\in D'\). \(\square\) A function \(f:\bbc\to F\) is called entire if it is holomorphic on \(\bbc\). From Theorem 25 we can see that a function is entire if and only if it is represented by a power series with infinite radius of convergence.

Theorem 26. Let \(f\) be an entire function. If there is a constant \(c\) and a positive integer \(k\) such that $$\sup_{|z|=r} |f(z)| \le cr^k$$ for all \(r > 0\), then \(f\) is a polynomial of degree \(k\) or less (with coefficients in \(F\)).

Proof. Write \(f(z)=\sum_{n=0}^\infty a_n z^n\) where \(a_n\in F\). By Theorem 25, we have $$
|a_n| \le \frac{1}{r^n} \sup_{|z|=r} |f(z)| \le cr^{k-n}.
$$ If \(n > k\), we can take \(r\to\infty\) to deduce that \(a_n=0\). \(\square\)

Corollary 27 (Liouville’s theorem). Any bounded entire function is constant.

As a simple application, we show that the spectrum of any element of a complex Banach algebra is nonempty. Let \(A\) be a complex unital Banach algebra and let \(x\in A\). We define the spectrum of \(x\), denoted by \(\sigma(x)\), to be the set of numbers \(\lambda\in\bbc\) such that \(x-\lambda 1\) is not invertible (where \(1\) is the unit in \(A\)).

Theorem 28. The spectrum of any \(x\in A\) is nonempty.

Proof. Suppose that \(\sigma(x)\) is empty. The map \(f:\bbc\to A\) given by \(z\mapsto (x-z1)^{-1}\) is entire, since \begin{align}
Df(z)w &= -(x-z1)^{-1}(-w1)(x-z1)^{-1} \\
&= w(x-z1)^{-2}.
\end{align} (See this result.) If \(|z| > |x|\) then \(x-z1=-z(1-x/z)\) is invertible and $$
|(x-z1)^{-1}|=|z|^{-1}|(1-x/z)^{-1}|\le\frac{|z|^{-1}}{1-|x/z|} \to 0
$$ as \(|z|\to\infty\). By Liouville’s theorem, \(f=0\). But \(f(z)=(x-z1)^{-1}\ne 0\) for any \(z\in\bbc\), which is a contradiction. \(\square\)

Cauchy’s theorem and winding numbers

An important consequence of Theorem 17 is the following result, which is the global version of Cauchy’s theorem (Theorem 23):

Theorem 29 (Cauchy’s theorem). Let \(U\subseteq\bbc\) be an open set, let \(\gamma_1,\gamma_2\) be 1-cycles in \(U\) that are homologous, and let \(f\) be holomorphic on \(U\). Then $$
\int_{\gamma_1} f = \int_{\gamma_2} f.
$$

Usually, Cauchy’s theorem is stated in terms of winding numbers (which will be defined shortly). Our goal is to prove the following:

Theorem 30 (Cauchy’s theorem with winding numbers). Let \(U\subseteq\bbc\) be an open set, let \(\gamma_1,\gamma_2\) be 1-cycles in \(U\) such that \(W(\gamma_1,z)=W(\gamma_2,z)\) for all \(z\in\bbc\setminus U\), and let \(f\) be holomorphic on \(U\). Then $$\int_{\gamma_1} f = \int_{\gamma_2} f.$$

We recall some concepts and theorems from algebraic topology. For any topological space \(X\), we define a loop in \(X\) to be a continuous map \(\gamma:[0,1]\to X\) with \(\gamma(0)=\gamma(1)\), and we say that \(\gamma\) is based at \(\gamma(0)\). Let \(\mathbb{S}^1=\{z\in\bbc:|z|=1\}\) be the circle. The map \(q:\mathbb{R}\to\mathbb{S}^1\) given by \(s\mapsto e^{2\pi is}\) is a universal covering of \(\mathbb{S}^1\). Let \(f:[0,1]\to\mathbb{S}^1\) be a loop based at a point \(z_0\in\mathbb{S}^1\). We define the winding number of \(f\) by \(\widetilde{f}(1)-\widetilde{f}(0)\), where \(\widetilde{f}:[0,1]\to\mathbb{R}\) is any lift of \(f\). Since any two lifts of \(f\) differ by a constant, the winding number is well-defined. Since \(\widetilde{f}(1)\) and \(\widetilde{f}(0)\) are both elements of the fiber \(q^{-1}(\{z_0\})\), they differ by an integer; thus the winding number of a loop is always an integer.

Theorem 31. Let \(f,g\) be loops in \(\mathbb{S}^1\) based at the same point. Then \(f\) and \(g\) are (path) homotopic if and only if they have the same winding number.

Now let \(z_0\in\bbc\) and let \(\gamma:[0,1]\to\bbc\setminus\{z_0\}\) be a closed path. Define a retraction \(r:\bbc\setminus\{z_0\}\to\mathbb{S}^1\) by $$r(z)=\frac{z-z_0}{|z-z_0|}.$$ Then \(r\circ\gamma\) is a loop in \(\mathbb{S}^1\), and we can define the winding number of \(\gamma\) with respect to \(z_0\) to be the winding number of \(r\circ\gamma\). We denote this integer by \(W(\gamma,z_0)\). If we consider \(\gamma\) as a singular 1-cycle in the homology group \(H_1(\bbc\setminus\{z_0\})\cong\mathbb{Z}\), then \([\gamma]=W(\gamma,z_0)[\alpha]\) where \(\alpha\) is the generator of \(H_1(\bbc\setminus\{z_0\})\) defined by \(\alpha(s)=z_0+e^{2\pi is}\). Therefore, for any 1-cycle \(\gamma\) in \(\bbc\setminus\{z_0\}\) we define the winding number of \(\gamma\) with respect to \(z_0\) to be the unique integer \(W(\gamma,z_0)\) such that \([\gamma]=W(\gamma,z_0)[\alpha]\).

Theorem 32.

  1. If \(\gamma\) is homologous to \(\eta\) in \(\bbc\setminus\{z_0\}\), then \(W(\gamma,z_0)=W(\eta,z_0)\).
  2. If \(\gamma_1,\dots,\gamma_k\) are closed paths and \(n_1,\dots,n_k\) are integers, then $$
    W(n_1\gamma_1+\cdots+n_k\gamma_k,z_0)=n_1 W(\gamma_1,z_0)+\cdots+n_k W(\gamma_k,z_0).
    $$

Note that we have a convenient expression for the winding number of a 1-cycle as an integral:

Theorem 33. For every 1-cycle \(\gamma\) in \(\bbc\setminus\{z_0\}\), we have $$
W(\gamma,z_0)=\frac{1}{2\pi i}\int_\gamma \frac{1}{z-z_0}\,dz.
$$

Proof. We first prove the result for closed paths in \(\bbc\setminus\{z_0\}\). By Theorem 12, we may assume that \(\gamma\) is a closed curve. By linearity, we may also assume that \(\gamma\) is a \(C^1\) path. Let \(\widetilde{\gamma}:[0,1]\to\mathbb{R}\) be a lift of \(r\circ\gamma\); then \(\widetilde{\gamma}\) is \(C^1\) and $$
e^{2\pi i\widetilde{\gamma}(s)} = \frac{\gamma(s)-z_0}{f(s)},
$$ where \(f(s)=|\gamma(s)-z_0|\). We compute \begin{align}
\frac{1}{2\pi i}\int_\gamma \frac{1}{z-z_0}\,dz &= \frac{1}{2\pi i}\int_0^1 \frac{\gamma'(s)}{\gamma(s)-z_0}\,ds \\
&= \frac{1}{2\pi i}\int_0^1 \frac{2\pi if(s)\widetilde{\gamma}'(s)e^{2\pi i\widetilde{\gamma}(s)}+f'(s)e^{2\pi i\widetilde{\gamma}(s)}}{f(s)e^{2\pi i\widetilde{\gamma}(s)}}\,ds \\
&= \frac{1}{2\pi i}\int_0^1 \left(2\pi i\widetilde{\gamma}'(s)+\frac{f'(s)}{f(s)}\right)\,ds \\
&= \frac{1}{2\pi i}[2\pi i\widetilde{\gamma}(s)+\log f(s)]_0^1 \\
&= \widetilde{\gamma}(1)-\widetilde{\gamma}(0) \\
&= W(\gamma,z_0).
\end{align} Now let \(\gamma\) be a 1-cycle in \(\bbc\setminus\{z_0\}\). By Theorem 15, \(\gamma\) is homologous to a sum \(\sum_{j=1}^k c_j \gamma_j\) where each \(\gamma_j\) is a closed path. Then $$
W(\gamma,z_0)=\sum_{j=1}^k c_j W(\gamma_j,z_0) = \sum_{j=1}^k c_j \frac{1}{2\pi i} \int_{\gamma_j} \frac{1}{z-z_0}\,dz = \frac{1}{2\pi i}\int_\gamma \frac{1}{z-z_0}\,dz.
$$ \(\square\)

Lemma 34. Let \(\gamma:[a,b]\to\bbc\) be a curve and let \(A=\gamma([a,b])\). The function $$\alpha\mapsto\int_\gamma \frac{1}{z-\alpha}\,dz$$ is continuous on \(\bbc\setminus A\).

Proof. Let \(\alpha_0\in\bbc\setminus A\). The function \(t\mapsto|\alpha_0-\gamma(t)|\) is positive and continuous on \([a,b]\), so it attains a minimum \(r > 0\). For all \(|\alpha-\alpha_0| < r/2\) and \(t\in[a,b]\) we have $$ |\alpha-\gamma(t)| \ge |\alpha_0-\gamma(t)|-|\alpha-\alpha_0| \ge r/2, $$ so \begin{align} \left\vert \int_\gamma\left(\frac{1}{z-\alpha}-\frac{1}{z-\alpha_0}\right)\,dz \right\vert &\le L(\gamma) \sup_{t\in[a,b]} \left\vert\frac{\alpha-\alpha_0}{(\gamma(t)-\alpha)(\gamma(t)-\alpha_0)}\right\vert \\ &\le L(\gamma)\frac{4}{r^2}|\alpha-\alpha_0| \\ &\to 0 \end{align} as \(\alpha\to\alpha_0\). \(\square\) Corollary 35. Let \(\gamma:[a,b]\to\bbc\) be a closed curve and let \(A=\gamma([a,b])\). If \(E\) is a connected subset of \(\bbc\setminus A\), then \(z\mapsto W(\gamma,z)\) is constant on \(E\). If \(E\) is unbounded, then \(W(\gamma,z)=0\) for all \(z\in E\).

Proof. The first claim is clear. Let \(n\) be the winding number of \(\gamma\) with respect to any point of \(E\). We have $$
n=\frac{1}{2\pi i}\int_\gamma\frac{1}{\zeta-z}\,d\zeta
$$ for all \(z\in E\), so \(n=0\) since $$
\left\vert \int_\gamma\frac{1}{\zeta-z}\,d\zeta \right\vert \to 0
$$ as \(|z|\to\infty\). \(\square\)

We now come to the fundamental theorem that links singular homology and winding numbers. We will provide a proof later.

Theorem 36. Let \(U\subseteq\bbc\) be an open set and let \(\gamma\) be a 1-cycle in \(U\). If \(W(\gamma,z)=0\) for all \(z\in\bbc\setminus U\), then \(\gamma\) is a boundary, i.e. \(\gamma=\partial b\) for some 2-chain \(b\).

Corollary 37. Let \(\gamma,\eta\) be 1-cycles in \(U\). Then \(\gamma\) and \(\eta\) are homologous if and only if \(W(\gamma,z)=W(\eta,z)\) for all \(z\in\bbc\setminus U\).

Clearly, Theorem 30 (our goal) follows directly from Corollary 37.

If \(c=\sum_{i=1}^k c_i\sigma_i\) is a \(p\)-chain where \(c_i\ne 0\), we define the image of \(c\) to be the set \(\bigcup_{i=1}^k \sigma_i(\triangle_p)\).

Theorem 38 (Cauchy’s integral formula). Let \(U\subseteq\bbc\) be an open set, let \(\gamma\) be a 1-cycle in \(U\) homologous to 0, and let \(f\) be holomorphic on \(U\). For all \(z\in U\) not in the image of \(\gamma\) we have $$
W(\gamma,z)f(z)=\frac{1}{2\pi i}\int_\gamma\frac{f(\zeta)}{\zeta-z}\,d\zeta.
$$

Proof. Write \(f(\zeta)=\sum_{n=0}^\infty a_n(\zeta-z)^n\) in a neighborhood of \(z\). Let \(C\) be a small circle centered at \(z\), contained in this neighborhood. By Theorem 30, \begin{align}
\frac{1}{2\pi i}\int_\gamma\frac{f(\zeta)}{\zeta-z}\,d\zeta &= \frac{1}{2\pi i}\int_{W(\gamma,z)C}\frac{f(\zeta)}{\zeta-z}\,d\zeta \\
&= \frac{1}{2\pi i}\sum_{n=0}^\infty \int_{W(\gamma,z)C} a_n(\zeta-z)^{n-1}\,d\zeta \\
&= a_0\frac{1}{2\pi i}\int_{W(\gamma,z)C} \frac{1}{\zeta-z}\,d\zeta \\
&= W(\gamma,z)f(z).
\end{align} \(\square\)

Proof of the theorem

If \(\gamma:[a,b]\to\bbc\) is a closed curve and there exists a partition \(\{a_0,\dots,a_k\}\) of \([0,1]\) such that \(\gamma|_{[a_{j-1},a_j]}\) is a horizontal or vertical line segment for each \(j\), then we say that \(\gamma\) is rectangular. A rectangular 1-cycle is a 1-cycle that can be written as a sum of rectangular closed curves. A grid is a union of finitely many vertical or horizontal lines in \(\bbc\). Every grid partitions \(\bbc\) into a finite number of rectangular regions, some bounded and some unbounded. Then it is clear that for any rectangular 1-cycle \(\gamma\) there is a grid \(G\) for which \(\gamma=\sum_{i=1}^k c_i\sigma_i\), where each \(\sigma_i\) is an edge of a bounded rectangle. We say that \(G\) is a grid for \(\gamma\).

Lemma 39. Let \(\gamma\) be a rectangular 1-cycle in \(\bbc\), let \(G\) be a grid for \(\gamma\), and let \(R_1,\dots,R_n\) be the bounded rectangles. For each \(i\), choose some \(p_i\in\Int R_i\). Then $$
\gamma=\sum_{i=1}^n W(\gamma,p_i)\partial R_i.
$$ (Each \(\partial R_i\) is oriented counterclockwise.)

Proof. Let \(\eta=\gamma-\sum_{i=1}^n W(\gamma,p_i)\partial R_i\); it is clear that \(W(\eta,p)=0\) for any \(p\) not on the grid (i.e. not on the boundary of some bounded or unbounded rectangle). Suppose that \(\eta\ne 0\) and write \(\eta=m\sigma+\eta’\), where \(m\ne 0\), \(\sigma\) is an edge of a bounded rectangle \(R\), and \(\eta’\) is some 1-chain not containing \(\sigma\). Then \(\sigma\) is also an edge of exactly one other rectangle \(R’\), which is either bounded or unbounded. Choose \(p\in\Int R\) and \(p’\in\Int R’\). Then \(W(\partial R,p)=1\) and \(W(\partial R,p’)=0\), so \begin{align}
W(\eta-m\partial R,p) &= W(\eta,p)-mW(\partial R,p)=-m, \\
W(\eta-m\partial R,p’) &= W(\eta,p’)-mW(\partial R,p’)=0.
\end{align} But the image \(E\) of \(\eta-m\partial R\) does not contain the edge \(\sigma\), so \(p\) and \(p’\) are in the same connected component of \(\bbc\setminus E\). Therefore $$
W(\eta-m\partial R,p)=W(\eta-m\partial R,p’)
$$ by Corollary 35, which is a contradiction. \(\square\)

Proof of Theorem 36. By using an argument similar to that of Theorem 12, we may assume that \(\gamma\) is a rectangular 1-cycle in \(U\). Let \(G\) be a grid for \(\gamma\), let \(R_1,\dots,R_n\) be the bounded rectangles, and choose some \(p_i\in\Int R_i\) for each \(i\). By Lemma 39, we have $$\gamma=\sum_{i=1}^n W(\gamma,p_i)\partial R_i.$$ Suppose some \(R_i\) contains a point \(p\in\bbc\setminus U\); then \(W(\gamma,p)=0\). If \(p\in\Int R_i\), then \(W(\gamma,p_i)=W(\gamma,p)=0\) since \(\Int R_i\) is connected. If \(p\in\partial R_i\) and \(p\) is not in the image of \(\gamma\), then again we have \(W(\gamma,p_i)=W(\gamma,p)=0\). Note that \(p\) cannot be in the image of \(\gamma\). Therefore \(R_i\subseteq U\) whenever \(W(\gamma,p_i)\ne 0\), and \(\gamma\) is the boundary of the 2-chain $$
\sum_{i=1}^n W(\gamma,p_i) R_i.
$$ \(\square\)

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