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Theorem 6.16. Let \( A \) denote the annulus \( {R}_{1} < \left| {z - c}\right| < {R}_{2} \) . If \( \gamma \) is a closed path in \( A \) and if \( f \) is holomorphic in \( A \) with Laurent series \( f\left( z\right) = \mathop{\sum }\limits_{{n = - \infty }}^{{+\infty }}{a}_{n}{\left( z - c\right) }^{n} \) , then\n\... | Proof. Write\n\n\[ \ng\left( z\right) = \mathop{\sum }\limits_{{n \neq - 1}}{a}_{n}{\left( z - c\right) }^{n}\text{ and }f\left( z\right) = \frac{{a}_{-1}}{z - c} + g\left( z\right) .\n\]\n\nThe function \( g \) has a primitive in the annulus \( A \) ; namely,\n\n\[ \n\mathop{\sum }\limits_{{n \neq - 1}}\frac{1}{n + 1}... | Yes |
Theorem 6.17 (Residue Theorem). Let \( f \) be holomorphic in a domain \( D \subseteq \mathbb{C} \) except for isolated singularities at \( {z}_{1},\ldots ,{z}_{n} \) in D. If \( \gamma \) is a positively oriented Jordan curve homotopic to a point in \( D \) and all \( {z}_{j} \) are in the interior of \( \gamma \), th... | Proof. Put a small positively oriented circle around each \( {z}_{j} \) and use the extended version of Cauchy's Theorems 5.26and 6.16. | No |
Theorem 6.18 (The Argument Principle). Let \( D \) be a domain in \( \mathbb{C} \) and let \( f \in \) \( \mathbf{M}\left( D\right) \) . Suppose \( \gamma \) is a positively oriented Jordan curve in \( D \) which is homotopic to a point in \( D \) . Let \( c \in \mathbb{C} \) and assume that \( f\left( z\right) \neq c ... | Proof. If \( F\left( z\right) = \frac{{f}^{\prime }\left( z\right) }{f\left( z\right) - c} \) for \( z \in D \), then \( F \in \mathbf{M}\left( D\right) \), and we claim that \( F \) has only simple poles, at the zeros and poles of \( f - c \), and that\n\n\[ \n\operatorname{Res}\left( {F\left( z\right) \mathrm{d}z, d}... | Yes |
Corollary 6.20. Let \( f \) be a nonconstant holomorphic function on a neighborhood of \( c \in \mathbb{C},\alpha = f\left( c\right) \), and \( m = {v}_{c}\left( {f - \alpha }\right) \) . Then there exist \( r > 0 \) and \( \varepsilon > 0 \) such that for all \( \beta \in \mathbb{C} \) with \( 0 < \left| {\beta - \alp... | Proof. Observe that \( m \geq 1 \) . Choose a positively oriented circle \( \gamma \) around \( c \) such that \( f - \alpha \) vanishes only at \( c \) in cl \( i\left( \gamma \right) \) and \( {f}^{\prime }\left( z\right) \neq 0 \) for all \( z \in \operatorname{cl}i\left( \gamma \right) - \{ c\} \) . If we consider ... | Yes |
Corollary 6.21. A nonconstant holomorphic function is an open mapping. | Proof. If \( f : D \rightarrow \mathbb{C} \) is holomorphic on a domain \( D \) and is not a constant, we obtain from Corollary 6.20 that for any \( \alpha \) in \( f\left( D\right) \) there exists \( \epsilon > 0 \) such that \( U\left( {\alpha ,\epsilon }\right) \subseteq f\left( D\right) \), and the result follows. | Yes |
Theorem 6.24 (Rouché's Theorem). Let \( f \) and \( g \) be holomorphic functions on a domain \( D \) . Let \( \gamma \) be a positively oriented Jordan curve with \( \operatorname{cl}i\left( \gamma \right) \) contained in \( D \) , and assume that \( \left| f\right| > \left| g\right| \) on range \( \gamma \) . Then \(... | Proof. It follows from Theorem 6.18 that \( {Z}_{f} = I\left( {f \circ \gamma ,0}\right) \) . Now apply Theorem 4.60 with \( {\gamma }_{1} = g \circ \gamma \) and \( {\gamma }_{2} = f \circ \gamma \) . | Yes |
Example 6.25. Rouché's theorem is very useful in locating zeros of a holomorphic function, as this example shows. Let \( h\left( z\right) = {z}^{5} + {z}^{4} + {6z} + 1 \) . Then | \[ \left| {{z}^{4} + {6z} + 1}\right| \leq {29} < {32} = \left| {z}^{5}\right| \] for all \( \left| z\right| = 2 \), and \[ \left| {{z}^{5} + {z}^{4} + 1}\right| \leq 3 < 6 = \left| {6z}\right| \] for all \( \left| z\right| = 1 \) . Therefore \( h \) has its five zeros contained in \( \left| z\right| < 2 \), and four o... | Yes |
Proposition 6.27. Let \( D \) be a domain in \( \mathbb{C},{z}_{0} \) a point in \( D \), and \( f \) a function holomorphic on \( D \). Then the following properties hold:\n\n(1) If \( {f}^{\prime }\left( {z}_{0}\right) \neq 0 \), then \( f \) defines a homeomorphism of some neighborhood of \( {z}_{0} \) onto some nei... | Proof. Let \( {w}_{0} = f\left( {z}_{0}\right) \). We proceed to establish the various parts of the theorem.\n\n(1) The condition implies that \( {v}_{{z}_{0}}\left( {f\left( z\right) - {w}_{0}}\right) = 1 \), and it follows from Corollary 6.20 that there exist \( r > 0 \) and \( \varepsilon > 0 \) such that for all \(... | Yes |
Let \( 0 < b < a \) and evaluate\n\n\[ I = {\int }_{0}^{2\pi }\frac{1}{a + b\cos \theta }\mathrm{d}\theta = {\int }_{\left| z\right| = 1}\frac{1}{\left( {\imath z}\right) \left( {a + b\left( {z + \frac{1}{z}}\right) \frac{1}{2}}\right) }\mathrm{d}z \] | \[ = {\int }_{\left| z\right| = 1}\frac{-2\imath }{b{z}^{2} + {2az} + b}\mathrm{\;d}z \]\n\n\[ = {2\pi }\imath \mathop{\sum }\limits_{{\left| z\right| < 1}}\operatorname{Res}\left( {\frac{-2\imath }{b{z}^{2} + {2az} + b}, z}\right) . \]\n\nThe denominator of the integrand in the last integral is a quadratic polynomial ... | Yes |
Lemma 6.33. Under the hypothesis of the last definition, there exists a constant \( M > 0 \) independent of \( \delta \) such that\n\n\[{\int }_{{\gamma }_{\delta }}\omega - {\int }_{{\gamma }_{d,\delta }}\omega = - \imath \mathop{\sum }\limits_{{k = 1}}^{m}{\alpha }_{\delta, k}\operatorname{Res}\left( {\omega ,{w}_{k}... | Proof. Only the first identity needs verification. Fix \( k \) . By a translation and rotation, we may assume that \( {w}_{k} = 0 \) and \( {a}_{k} \in {\mathbb{R}}_{ < 0} \) . Write\n\n\[ f\left( z\right) = \frac{\rho }{z} + g\left( z\right) \text{ for }\left| z\right| < \delta ,\]\n\nwith \( {M}_{0} \) a bound for \(... | Yes |
Theorem 6.34 (Residue Theorem, Version 2). Let \( \gamma \) be a closed positively oriented Jordan curve in a domain \( D \subset \mathbb{C} \) with \( i\left( \gamma \right) \subset D \) . Let \( f \) be a holomorphic function in the domain \( D \) except for isolated singularities at \( {z}_{1},\ldots ,{z}_{n} \) in ... | Proof. As in previous arguments, \( \delta \) is a small positive number, but in this case at most the minimum of the distances from any of the \( {w}_{k} \) to the \( {z}_{j} \) . We will use much of the notation previously introduced in this section. By introducing new line segments and thus integrating over a finite... | Yes |
A necessary and sufficient condition for a sequence of functions \( \\left\\{ {f}_{n}\\right\\} \\subset \\mathbf{C}\\left( D\\right) \) to converge uniformly on all compact subsets of \( D \) is for the sequence to converge uniformly on all compact discs with rational centers and rational radii contained in \( D \) . | Proof. Every compact set contained in \( D \) can be covered by finitely many such discs. | Yes |
Theorem 7.2. If \( \left\{ {f}_{n}\right\} \subset \mathbf{H}\left( D\right) \) and \( \left\{ {f}_{n}\right\} \) converges uniformly on all compact subsets of \( D \), then the limit function \( f \) is holomorphic on \( D \) . | Proof. We already know that \( f \in \mathbf{C}\left( D\right) \) . Let \( \gamma \) be any closed curve homotopic to a point in \( D \) . Then, by Cauchy’s theorem,\n\n\[ \n{\int }_{\gamma }{f}_{n}\left( z\right) \mathrm{d}z = 0 \n\]\n\nBy uniform convergence it follows that\n\n\[ \n{\int }_{\gamma }f\left( z\right) \... | Yes |
Theorem 7.4. If \( \left\{ {f}_{n}\right\} \subset \mathbf{H}\left( D\right) \) and \( {f}_{n} \rightarrow f \) uniformly on all compact subsets of \( D \), then \( {f}_{n}^{\prime } \rightarrow {f}^{\prime } \) uniformly on all compact subsets of \( D \) . | Proof. Since \( f \in \mathbf{H}\left( D\right) \), it is enough to check uniform convergence of the derivatives on all compact subdiscs \( R \subset D \), with \( \partial R = \gamma \) positively oriented. For \( z \in i\left( \gamma \right) \), we have\n\n\[ \n{f}^{\prime }\left( z\right) = \frac{1}{{2\pi }\imath }{... | Yes |
Theorem 7.5. Let \( \\left\\{ {f}_{n}\\right\\} \) be a sequence of holomorphic functions on \( D \) such that \( {f}_{n} \\rightarrow f \) uniformly on all compact subsets of \( D \) . If \( {f}_{n}\\left( z\\right) \\neq 0 \) for all \( z \\in D \) and all \( n \\in {\\mathbb{Z}}_{ > 0} \), then either\n\n(a) \( f \)... | Proof. Assume that there is \( c \\in D \) with \( f\\left( c\\right) = 0 \) and that \( f \) is not identically zero. Then there exists a circle \( \\gamma \) with center \( c \) such that \( \\operatorname{cl}i\\left( \\gamma \\right) \\subset D \) and \( f\\left( z\\right) \\neq 0 \) for all \( z \\in \\operatorname... | Yes |
Theorem 7.9 (Hurwitz). Assume \( D \) is a domain in \( \mathbb{C} \). If \( \left\{ {f}_{n}\right\} \) is a sequence in \( \mathbf{H}\left( D\right) \) with \( {f}_{n} \rightarrow f \) uniformly on all compact subsets of \( D \) and \( {f}_{n} \) is schlicht for each \( n \), then either \( f \) is constant or schlich... | Proof. Assume that \( f \) is neither constant nor schlicht; thus, in particular, there exist \( {z}_{1} \) and \( {z}_{2} \) in \( D \) with \( {z}_{1} \neq {z}_{2} \) and \( f\left( {z}_{1}\right) = f\left( {z}_{2}\right) \). For each \( n \in {\mathbb{Z}}_{ > 0} \), set \( {g}_{n}\left( z\right) = \) \( {f}_{n}\left... | Yes |
Theorem 7.11. Convergence in the \( \rho \) -metric in \( \mathbf{C}\left( D\right) \) is equivalent to uniform convergence on all compact subsets of \( D \) . | Proof. Let \( \left\{ {f}_{n}\right\} \subset \mathbf{C}\left( D\right) \) and assume that \( \left\{ {f}_{n}\right\} \) is \( \rho \) -convergent. Since for every compact set \( K \subset D \) there is an \( i \) in \( {\mathbb{Z}}_{ > 0} \) such that \( K \subset {K}_{i} \), it suffices to show uniform convergence on... | Yes |
Corollary 7.13. \( \rho \) is a complete metric on \( \mathbf{C}\left( D\right) \) . | Because of Theorem 7.11, we can reformulate the results of the previous section in terms of the metric \( \rho \) . In particular, Theorems 7.2 and 7.4 can now be phrased as in the following corollary. We already remarked that \( \mathbf{H}\left( D\right) \subset \mathbf{C}\left( D\right) \) . We let \( {\left. \rho \r... | No |
Theorem 7.17. For any \( f \in \mathbf{C}\left( D\right) \), a basis for the neighborhood system at \( f \) (with respect to the topology induced by \( \rho \) on \( \mathbf{C}\left( D\right) \) ) is given by the sets \( {V}_{f}\left( {K,\epsilon }\right) \) . | Proof. It is enough to show that\n\n(1) Given \( {V}_{f}\left( {K,\epsilon }\right) \), there exists an \( {N}_{f}\left( \delta \right) \subseteq {V}_{f}\left( {K,\epsilon }\right) \),\n\nand\n\n(2) Given \( {N}_{f}\left( \delta \right) \), there exists a \( {V}_{f}\left( {K,\epsilon }\right) \subseteq {N}_{f}\left( \d... | Yes |
Theorem 7.20. Let \( \left\{ {f}_{n}\right\} \subset \mathbf{M}\left( D\right) \) . If \( \sum {f}_{n} \) converges uniformly on all compact subsets of \( D \), then the series \( f = \sum {f}_{n} \) is a meromorphic function on \( D \), and \( \sum {f}_{n}^{\prime } \) converges uniformly on all compact subsets to \( ... | Proof. The proof is trivial. | No |
Theorem 7.21. For all \( z \) in \( \mathbb{C} - \mathbb{Z} \) the following equalities hold:\n\n\[ \pi \cot {\pi z} = \frac{\pi \cos {\pi z}}{\sin {\pi z}} = \frac{1}{z} + \mathop{\sum }\limits_{{n \in \mathbb{Z}, n \neq 0}}\left\lbrack {\frac{1}{z - n} + \frac{1}{n}}\right\rbrack \]\n\n\[ = \frac{1}{z} + {2z}\mathop{... | Proof of Theorem 7.21. For \( N \in {\mathbb{Z}}_{ > 0} \), let \( {C}_{N} \) be the positively oriented boundary of the square with vertices \( \left( {N + \frac{1}{2}}\right) \left( {\pm 1 \pm \iota }\right) \) ; see Fig. 7.1.\n\nThen\n\n\[ \frac{1}{{2\pi }\imath }{\int }_{{C}_{N}}\frac{\cot {\pi t}}{t - z}\mathrm{\;... | No |
Corollary 7.24. For all \( z \in \mathbb{C} - \mathbb{Z} \) , \[ \frac{{\pi }^{2}}{{\sin }^{2}{\pi z}} = \mathop{\sum }\limits_{{n = - \infty }}^{\infty }\frac{1}{{\left( z - n\right) }^{2}}. \] | In particular, setting \( z = \frac{1}{2} \) , \[ \frac{{\pi }^{2}}{4} = \mathop{\sum }\limits_{{n = - \infty }}^{\infty }\frac{1}{{\left( 2n - 1\right) }^{2}} \] | Yes |
Lemma 7.27. A set \( A \subseteq \mathbf{C}\left( D\right) \) is strongly bounded if and only if for each compact \( K \subset D \), there exists an \( M\left( K\right) > 0 \) such that \( \parallel f{\parallel }_{K} \leq M\left( K\right) \) for all \( f \in A \) ; that is, \( A \) is strongly bounded if and only if th... | Proof. We leave the proof as Exercise 7.6. | No |
Theorem 7.28. A compact subset \( A \subseteq \mathbf{C}\left( D\right) \) is closed and strongly bounded. | Proof. Since \( A \) is a compact subset of the metric space \( \mathbf{C}\left( D\right) \), it is closed. If \( K \subset D \) is compact, then the function\n\n\[ f \mapsto \parallel f{\parallel }_{K} \]\n\n is continuous on the compact set \( A \) . Thus it is a bounded function, and hence \( A \) is strongly bounde... | Yes |
Lemma 7.29. Let \( c \in \mathbb{C} \) and \( D = U\left( {c, R}\right) \) for some \( R > 0 \) . Assume \( A \subseteq \mathbf{H}\left( D\right) \) is strongly bounded, and let \( {\left\{ {f}_{k}\right\} }_{k \in {\mathbb{Z}}_{ > }0} \subseteq A \) . The sequence \( \left\{ {f}_{k}\right\} \) converges uniformly on a... | Proof. If \( {f}_{k} \rightarrow f \) uniformly on all compact subsets of \( D \), then for every nonnegative integer \( n,{f}_{k}^{\left( n\right) } \rightarrow {f}^{\left( n\right) } \) uniformly on all compact subsets of \( D \) ; in particular, \( {f}_{k}^{\left( n\right) }\left( c\right) \rightarrow {f}^{\left( n\... | Yes |
Theorem 7.30 (Compactness Theorem). Let \( D \) be a domain in \( \mathbb{C} \). Then every closed subset \( A \) of \( \mathbf{H}\left( D\right) \) that is bounded in the strong sense is compact. | Proof. Cover \( D \) by countably many open discs \( {\left\{ U\left( {z}_{i},{r}_{i}\right) \right\} }_{i \in {\mathbb{Z}}_{ > 0}} \) whose closures are contained in \( D \). For each \( i \in {\mathbb{Z}}_{ > 0} \) and each \( n \in {\mathbb{Z}}_{ \geq 0} \), consider the mapping\n\n\[ \n{\lambda }_{i}^{n} : \mathbf{... | No |
Theorem 7.36 (Vitali’s Theorem). Let \( D \) be a domain in \( \mathbb{C} \), and assume that the elements in a sequence \( {\left\{ {f}_{k}\right\} }_{k \in {\mathbb{Z}}_{ > 0}} \subset \mathbf{H}\left( D\right) \) are uniformly bounded on compact subsets of \( D \) . Let \( S \subseteq D \) and assume that \( S \) ha... | Proof. The assumptions imply that the set \( A = {\left\{ {f}_{k}\right\} }_{k \in {\mathbb{Z}}_{ > 0}} \subset \mathbf{H}\left( D\right) \) is strongly bounded. It follows from Montel's theorem that its closure is compact, and therefore every subsequence of the sequence \( \left\{ {f}_{k}\right\} \) has a converging s... | Yes |
Lemma 7.39. Let \( U \) and \( V \) be open subsets of \( \mathbb{C} \) with \( V \subseteq U \) and \( \partial V \cap U = \varnothing \) . If \( H \) is a connected component of \( U \) and \( H \cap V \neq \varnothing \), then \( H \subseteq V \) . | Proof. Let \( c \) be a point in \( H \cap V \) and let \( G \) be the connected component of \( V \) containing \( c \) . It is enough to show that \( G = H \) . Now \( H \cup G \) is connected, contained in \( U \) and contains \( c \) . Since \( H \) is the component of \( U \), containing \( c, G \subseteq H \) . F... | Yes |
For every \( c \in \mathbb{C} - K \), the rational function\n\n\[ g : z \mapsto {\left( z - c\right) }^{-1} \in B\left( {K, S}\right) . \] | Proof. The proof consists of several claims that will need verification.\n\n- Let us choose \( R \in {\mathbb{R}}_{ > 0} \) so that \( K \subset U\left( {0, R}\right) \) . Let \( {z}_{0} \in \mathbb{C} \) be arbitrary with \( \left| {z}_{0}\right| > R \) and let \( h \) be a rational function with a pole only at \( {z}... | No |
Proposition 8.8. Given three distinct points \( {z}_{2},{z}_{3},{z}_{4} \) in \( \widehat{\mathbb{C}} \), there exists a unique Möbius transformation \( S \) with \( S\left( {z}_{2}\right) = 1, S\left( {z}_{3}\right) = 0 \), and \( S\left( {z}_{4}\right) = \infty \) . | Proof. The proof has two parts.\n\nUniqueness: If \( {S}_{1} \) and \( {S}_{2} \) are Möbius transformations that solve our problem, then \( {S}_{1} \circ {S}_{2}^{-1} \) is a Möbius transformation that fixes 1,0 and \( \infty \) and hence, by Proposition 8.7, it is the identity map.\n\nExistence: If the \( {z}_{i} \) ... | Yes |
Proposition 8.11. If \( {z}_{1},{z}_{2},{z}_{3},{z}_{4} \) are four distinct points in \( \widehat{\mathbb{C}} \), and \( T \) is any Möbius transformation, then\n\n\[ \left( {T\left( {z}_{1}\right), T\left( {z}_{2}\right), T\left( {z}_{3}\right), T\left( {z}_{4}\right) }\right) = \left( {{z}_{1},{z}_{2},{z}_{3},{z}_{4... | Proof. If we define \( S\left( z\right) = \left( {z,{z}_{2},{z}_{3},{z}_{4}}\right) \) for \( z \in \mathbb{C} - \{ 0,1\} \), then \( S \circ {T}^{-1} \) is a Möbius transformation taking \( T\left( {z}_{2}\right) \) to \( 1, T\left( {z}_{3}\right) \) to 0 and \( T\left( {z}_{4}\right) \) to \( \infty \) . Therefore\n\... | Yes |
Proposition 8.13. The cross ratio of four distinct points in \( \widehat{\mathbb{C}} \) is a real number if and only if the four points lie on a circle in \( \widehat{\mathbb{C}} \) . | Proof. This is an elementary geometric argument that goes as follows. It is clear that\n\n\[ \arg \left( {{z}_{1},{z}_{2},{z}_{3},{z}_{4}}\right) = \arg \frac{{z}_{1} - {z}_{3}}{{z}_{1} - {z}_{4}} - \arg \frac{{z}_{2} - {z}_{3}}{{z}_{2} - {z}_{4}}.\]\n\nIt is also clear from the geometry of the situation (see Fig. 8.1 ... | No |
Theorem 8.14. A Möbius transformation maps circles in \( \widehat{\mathbb{C}} \) to circles in \( \widehat{\mathbb{C}} \) . | Proof. This follows immediately from Propositions 8.11 and 8.13. | Yes |
Corollary 8.15. If \( w\left( z\right) = \frac{z - \iota }{z + \iota } \) for \( z \in {\mathbb{H}}^{2} \), then \( w \) is a conformal map of \( {\mathbb{H}}^{2} \) onto \( \mathbb{D} \) . | Proof. All Möbius transformations, in particular \( w \), are conformal. A calculation shows that \( w \) maps \( \widehat{\mathbb{R}} = \mathbb{R} \cup \{ \infty \} \) onto \( {S}^{1} \) (the unit circle centered at 0 ) and \( w\left( \iota \right) = \) 0 . By connectivity considerations, it follows that \( w\left( {\... | Yes |
Theorem 8.16. A function \( f : \mathbb{C} \rightarrow \mathbb{C} \) belongs to \( \operatorname{Aut}\left( \mathbb{C}\right) \) if and only if there exist \( a \) and \( b \) in \( \mathbb{C}, a \neq 0 \), such that \( f\left( z\right) = {az} + b \) for all \( z \in \mathbb{C} \) . | Proof. The if part is trivial. For the only if part, note that \( f \) is an entire function, and we can use its Taylor series at zero to conclude that\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{z}^{n}\text{ for all }z \in \mathbb{C}. \]\n\nIf \( \infty \) were an essential singularity of... | Yes |
Theorem 8.17. Aut \( \left( \widehat{\mathbb{C}}\right) \cong \operatorname{PSL}\left( {2,\mathbb{C}}\right) \) . Thus the last arrow in the exact sequence (8.3) corresponds to a surjective map. | Proof. We need only show that \( \operatorname{Aut}\left( \widehat{\mathbb{C}}\right) \) is contained in the Möbius group. Let \( f \) be an element of \( \operatorname{Aut}\left( \widehat{\mathbb{C}}\right) \) . If \( f\left( \infty \right) = \infty \), then \( f \) is a Möbius transformation by Theorem 8.16. If \( f\... | Yes |
Theorem 8.18. A function \( B \) defined on \( \mathbb{D} \) is in \( \operatorname{Aut}\left( \mathbb{D}\right) \) if and only if there exist \( a \) and \( b \) in \( \mathbb{C} \) such that \( {\left| a\right| }^{2} - {\left| b\right| }^{2} = 1 \) and \[ B\left( z\right) = \frac{{az} + b}{\bar{b}z + \bar{a}} \] for ... | Proof. The if part: Assume that \( B \) is of the above form and observe that \( a \) is different from zero. We show that \( B \in \operatorname{Aut}\left( \mathbb{D}\right) \) . This follows from the following easy to prove facts: (1) Mappings \( B \) of the given form constitute a group under composition. In particu... | Yes |
Theorem 8.19. Aut \( \left( {\mathbb{H}}^{2}\right) \cong \operatorname{PSL}\left( {2,\mathbb{R}}\right) \) . | Proof. Consider the conformal map \( w : {\mathbb{H}}^{2} \rightarrow \mathbb{D} \) given in Corollary 8.15. Then\n\n\[ \operatorname{Aut}\left( {\mathbb{H}}^{2}\right) = {w}^{-1}\operatorname{Aut}\left( \mathbb{D}\right) w. \]\n\nBy the preceding theorem, any element \( f \) of \( \operatorname{Aut}\left( \mathbb{D}\r... | Yes |
Corollary 8.21. If \( D \) is a nonempty simply connected domain in \( \widehat{\mathbb{C}} \), then \( D \) is conformally equivalent to one and only one of the following domains: (i) \( \widehat{\mathbb{C}} \) ,(ii) \( \mathbb{C} \), or (iii) \( \mathbb{D} \) . | Proof. Existence: If \( D \subsetneqq \widehat{\mathbb{C}} \), we may first reduce to the case \( D \subseteq \mathbb{C} \) by observing that if \( D \) contains \( \infty \), we can choose \( c \in \mathbb{C} - D \), and setting \( F\left( z\right) = \frac{1}{z - c} \) we have that \( F\left( D\right) \subseteq \mathb... | Yes |
Proposition 8.24. For every conformal map \( f \) defined on \( D \) ,\n\n\[{\lambda }_{f\left( D\right) }\left( {f\left( z\right) }\right) \left| {{f}^{\prime }\left( z\right) }\right| = {\lambda }_{D}\left( z\right) \text{ for all }z \in D.\] | Proof. If \( \pi \) is a Riemann map (for \( D \) ), so is \( \pi \circ {f}^{-1} \) (for \( f\left( D\right) \) ).\n\nAny infinitesimal metric on \( D \) allows us to define lengths of paths in \( D \), and hence a distance function on the domain. We work, of course, with the length element\n\n\[ \mathrm{d}s = {\lambda... | No |
Lemma 8.27. For every pair \( z \) and \( w \) of distinct points in \( {\mathbb{H}}^{2} \), there exists a unique circle centered at the real axis passing through them, and a unique geodesic in \( {\mathbb{H}}^{2} \) passing through them. | Proof. If \( \Re \left( z\right) = \Re \left( w\right) \), take \( \widetilde{C} \) to be the Euclidean line through \( z \) and \( w \) . Otherwise, let \( L \) be the perpendicular bisector of the Euclidean line segment connecting \( z \) and \( w \) . If \( c \) is the point where \( L \) intersects the real line, w... | No |
Lemma 8.29. Let \( P \) and \( Q \) be two points in \( {\mathbb{H}}^{2} \) lying on an Euclidean circle \( C \) centered on the real axis, and let \( \gamma \) be the arc of \( C \) in \( {\mathbb{H}}^{2} \) between \( P \) and \( Q \) . Assume further that the radii from the center of \( C \) to \( P \) and \( Q \) m... | Proof. Assume the circle \( C \) has radius \( r \) and is centered at \( c \) (see Fig. 8.3). Let \( z = \left( {x, y}\right) \) be an arbitrary point on \( \gamma \) and let \( t \) be the angle that the radius from \( z \) to the center of \( C \) makes with the positive real axis; then \( x = c + r\cos t \) and\n\n... | Yes |
Lemma 8.32. Let \( z \) and \( w \) distinct points in \( {\mathbb{H}}^{2} \). There exists a unique \( T \in \operatorname{Aut}\left( {\mathbb{H}}^{2}\right) \) such that \( T\left( {z}^{ * }\right) = 0, T\left( z\right) = \iota, T\left( w\right) = {\iota yw} \) with \( y > 1 \), and \( T\left( {w}^{ * }\right) = \inf... | Proof. Consider the unique circle \( \widetilde{C} \) centered on the real axis and passing through \( z \) and \( w \). Since the Möbius group is triply transitive, there exists a unique Möbius transformation \( T \) that maps \( {z}^{ * }, z,{w}^{ * } \) to \( 0,\iota ,\infty \), respectively. Since Möbius transforma... | Yes |
Lemma 8.33. If \( z \) and \( w \) are two distinct points in \( {\mathbb{H}}^{2} \), then the hyperbolic length of the geodesic segment \( \widetilde{\gamma } \) joining them is shorter than the hyperbolic length of any other pdp \( \gamma \) in \( {\mathbb{H}}^{2} \) joining them. | Proof. Write \( z = {x}_{z} + \imath {y}_{z} \) and \( w = {x}_{w} + \imath {y}_{w} \) .\n\nFirst consider the case \( {x}_{z} = {x}_{w} \) . Assume the curve \( \gamma \) is parameterized by the closed interval \( \left\lbrack {a, b}\right\rbrack \subset \mathbb{R} \) and \( \gamma \left( t\right) = x\left( t\right) +... | Yes |
Theorem 8.34. For any two distinct points \( z \) and \( w \) in \( {\mathbb{H}}^{2} \), the geodesic segment joining \( z \) to \( w \) is the unique curve that achieves the infimum defined by (8.9). | Using cross ratios to simplify notation, a routine computation establishes | No |
Proposition 8.35. For any two distinct points \( z \) and \( w \) in \( {\mathbb{H}}^{2} \), the hyperbolic distance between \( z \) and \( w \) is equal to the length of the geodesic segment \( \gamma \) joining \( z \) and \( w \), and is given by\n\n\[ \n{\rho }_{{\mathbb{H}}^{2}}\left( {z, w}\right) = {l}_{{\mathbb... | Proof. A fractional linear transformation preserves the cross ratio of any four points, and fractional linear transformations mapping \( {\mathbb{H}}^{2} \) to itself are isometries for the hyperbolic metric. This yields all the equalities except for the last one, which is a calculation (see Exercise 8.15). | No |
Proposition 8.37. An orientation preserving isometry \( f \) of \( \left( {{\mathbb{H}}^{2},{\rho }_{{\mathbb{H}}^{2}}}\right) \) that fixes the imaginary axis pointwise is the identity map. | Proof. Let \( z = x + \imath y \) and \( f = u + \imath v \) . For all positive real numbers \( t \) we have\n\n\[ \n{\rho }_{{\mathbb{H}}^{2}}\left( {z,{\iota t}}\right) = {\rho }_{{\mathbb{H}}^{2}}\left( {f\left( z\right), f\left( {\iota t}\right) }\right) = {\rho }_{{\mathbb{H}}^{2}}\left( {u\left( z\right) + {\iota... | Yes |
Theorem 8.38. The set of orientation-preserving isometries of \( {\mathbb{H}}^{2} \) with respect to the hyperbolic metric is precisely the set of fractional linear transformations mapping \( {\mathbb{H}}^{2} \) to itself; that is, \( \operatorname{PSL}\left( {2,\mathbb{R}}\right) \) . | Proof. If \( g \) is such an isometry of \( {\mathbb{H}}^{2} \), it preserves geodesics. Thus there is a fractional linear transformation \( f \) that preserves \( {\mathbb{H}}^{2} \) and such that \( f \circ g \) leaves invariant the imaginary axis. Following this map by an isometry of the form \( z \mapsto {kz} \) wi... | Yes |
Proposition 8.40. For all \( z \) and \( w \in \mathbb{D} \) , \n\n\[ \n{\rho }_{\mathbb{D}}\left( {w, z}\right) = \log \frac{\left| {1 - w\bar{z}}\right| + \left| {w - z}\right| }{\left| {1 - w\bar{z}}\right| - \left| {w - z}\right| }. \n\] \n\n(8.12) \n\nIn particular, \n\n\[ \n{\rho }_{\mathbb{D}}\left( {0, z}\right... | Proof. See Exercise 8.16. | No |
Proposition 8.45. If \( \mathbb{A} = \left\{ {{a}_{0},{a}_{1},\ldots ,{a}_{n}}\right\} \) is a nonempty finite sequence of points in \( \mathbb{D} \), then\n\n(a) \( B = {B}_{\mathbb{A}} \) is a meromorphic function on \( \mathbb{C} \cup \{ \infty \} \), with zeros precisely \( {}^{5} \) at the\n\n\[ n + 1\text{points}... | --- \n\n\( {}^{5} \) If \( {a}_{i} \) appears \( v \) times in our list \( \mathbb{A} \), then\n\n\[ {v}_{{a}_{i}}\left( B\right) = v\text{ and }{v}_{\frac{1}{{a}_{i}}}\left( B\right) = - v. \] | Yes |
Proposition 8.46. Let \( \mathbb{A} = \left\{ {{a}_{0},{a}_{1},\ldots ,{a}_{n}}\right\} \) be a nonempty finite sequence of points in \( \mathbb{D} \), and let \( T \) be any element of \( \operatorname{Aut}\left( \mathbb{D}\right) \) . Then\n\n\[ \n{B}_{\mathbb{A}} \circ T = \lambda {B}_{{T}^{-1}\left( \mathbb{A}\righ... | Proof. Since \( T \) belongs to \( \operatorname{Aut}\left( \mathbb{D}\right) \), there exist complex numbers \( a \) and \( b \), with \( {\left| a\right| }^{2} - \) \( {\left| b\right| }^{2} = 1 \), such that \( T\left( z\right) = \frac{{az} + b}{\bar{b}z + \bar{a}} \) for all \( z \) in \( \mathbb{D} \) . It suffice... | Yes |
Theorem 8.47. Let \( f \) be a holomorphic self-map of the open unit disc \( \mathbb{D},\mathbb{A} = \) \( \left\{ {{a}_{0},{a}_{1},\ldots ,{a}_{n}}\right\} \) a nonempty finite collection of points in \( \mathbb{D} \) and \( B = {B}_{\mathbb{A}} \) . Assume that \( f\left( {a}_{i}\right) = 0 \) for each \( {a}_{i} \) ... | Proof. To prove (a), it suffices to consider the special case with \( {a}_{0} = 0 \) . To verify this claim, let \( T \) be an automorphism of \( \mathbb{D} \) that sends 0 to \( {a}_{0} \) ; note that if we have the first inequality in (a) for \( f \circ T \) and \( {B}_{{T}^{-1}\left( \mathbb{A}\right) } \), then, as... | Yes |
Corollary 8.48 (Schwarz-Pick Lemma). Let \( f \) be a holomorphic self-map of the open unit disc \( \mathbb{D} \). If \( a \in \mathbb{D} \), then\n\n\[ \left| \frac{f\left( z\right) - f\left( a\right) }{1 - \overline{f\left( a\right) }f\left( z\right) }\right| \leq \left| \frac{z - a}{1 - \bar{a}z}\right| \;\text{ for... | Proof. Apply the previous theorem to the function \( {B}_{f\left( a\right) } \circ f \) . | No |
Example 9.3. \( \log \left| z\right| \) is harmonic on \( D = {\mathbb{C}}_{ \neq 0} \) | since it belongs to \( {\mathbf{C}}^{2}\left( D\right) \), and it is locally the real part of \( \log z \), a multi-valued but holomorphic function in \( D \) . | No |
Proposition 9.4. If \( g \) is real-valued and harmonic in \( D \), then it is locally the real part of an analytic function. The analytic function is unique up to an additive constant. | Proof. Let \( {D}^{\prime } \subseteq D \) be a simply connected region. Since \( g \) is harmonic in \( {D}^{\prime } \), it follows from (2) above that \( 2{g}_{z}\mathrm{\;d}z \) is closed on \( {D}^{\prime } \), and hence an exact form on \( {D}^{\prime } \) . Choose a holomorphic function \( f \) on \( {D}^{\prime... | Yes |
Corollary 9.7. Harmonic functions have the MVP, and hence they satisfy the maximum modulus principle. Real-valued harmonic functions also satisfy the maximum and minimum principles. | Proof. See Definition 5.28 and the properties that follow thereof. | No |
Proposition 9.9 (The Poisson Formula). If \( g \) is a harmonic function on the domain \( \left| z\right| < \rho \) for some \( \rho > 0 \), then, for each \( 0 < r < \rho \) , \[ g\left( z\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }g\left( {r{\mathrm{e}}^{i\theta }}\right) \cdot \frac{{r}^{2} - {\left| z\right| }^{2}}... | Proof. It suffices to assume that \( g \) is real-valued. To establish this formula we can thus apply Proposition 9.4 and choose the holomorphic function \( f \) on this domain with \( \Re f = g \) and \( g\left( 0\right) = f\left( 0\right) \), noting that there is a unique such \( f \) . The function \( f \) has a pow... | Yes |
Proposition 9.11. If \( g \) is harmonic and real-valued in \( \left| z\right| < \rho \) for some \( \rho > 0 \), then the harmonic conjugate of \( g \) vanishing at the origin is given by | \[ \frac{1}{{2\pi }\imath }{\int }_{0}^{2\pi }g\left( {r{\mathrm{e}}^{\iota \theta }}\right) \cdot \frac{r{\mathrm{e}}^{-{\iota \theta }}z - r{\mathrm{e}}^{\iota \theta }\bar{z}}{{\left| r{\mathrm{e}}^{\iota \theta } - z\right| }^{2}}\mathrm{\;d}\theta ,\;\text{ for }\left| z\right| < r < \rho . \] | Yes |
Theorem 9.12 (Harnack’s Inequalities). If \( g \) is a positive harmonic function on \( \left| z\right| < r \) that is continuous on \( \left| z\right| \leq r \), then\n\n\[ \frac{r - \left| z\right| }{r + \left| z\right| } \cdot g\left( 0\right) \leq g\left( z\right) \leq \frac{r + \left| z\right| }{r - \left| z\right... | Proof. Our starting point is (9.3). We use elementary estimates for the Poisson kernel:\n\n\[ \frac{r - \left| z\right| }{r + \left| z\right| } = \frac{{r}^{2} - {\left| z\right| }^{2}}{{\left( r + \left| z\right| \right) }^{2}} \leq \frac{{r}^{2} - {\left| z\right| }^{2}}{{\left| r{\mathrm{e}}^{t\theta } - z\right| }^... | Yes |
Theorem 9.14 (Harnack's Convergence Theorem). Let \( D \) be a domain and let \( \left\{ {u}_{j}\right\} \) be a nondecreasing sequence of real-valued harmonic functions on \( D \) . Then\n\n(a) Either \( \mathop{\lim }\limits_{{j \rightarrow \infty }}{u}_{j}\left( z\right) = + \infty \) for all \( z \in D \)\n\n(b) Th... | Proof. Since a nondecreasing sequence of real numbers converges if and only if it is bounded, the assumption that \( \mathop{\lim }\limits_{{j \rightarrow \infty }}{u}_{j}\left( z\right) \) is not \( + \infty \) for all \( z \in D \) allows us to conclude that there exist \( {z}_{0} \) in \( D \) and a real number \( M... | Yes |
Theorem 9.17. If \( {u}_{1} \) and \( {u}_{2} \) are harmonic functions on \( D \), then\n\n\[ \n{u}_{1}{}^{ * }\mathrm{\;d}{u}_{2} - {u}_{2}{}^{ * }\mathrm{\;d}{u}_{1}\n\]\n\nis a closed form on \( D \) . | Proof. To establish this assertion, it involves no loss of generality to assume that the functions are real-valued (see Exercise 9.4), and hence we may also assume (because the issue is local) that each function \( {u}_{j} \) has a single-valued harmonic conjugate \( {v}_{j} \) ; thus\n\n\[ \n{u}_{1}{}^{ * }\mathrm{\;d... | No |
Theorem 9.20. A continuous complex-valued function that satisfies the MVP is harmonic. | Proof. Let \( f \) be a continuous function on a domain \( D \), let \( c \in D \) and let \( {r}_{0} > 0 \) be sufficiently small so that \( \operatorname{cl}U\left( {c,{r}_{0}}\right) \subset D \) and \( f \) satisfies (5.7) for all \( r \leq {r}_{0} \). It suffices to assume that \( f \) is real-valued. Let \( v \) ... | Yes |
Theorem 9.22. Let \( \Omega \) be a nonempty region in the complex plane that is symmetric about the real axis.\n\nIf \( v \) is a real-valued and continuous function on \( {\Omega }^{ + } \cup \sigma \), and it is harmonic on \( {\Omega }^{ + } \) and zero on \( \sigma \), then \( v \) has a harmonic extension to \( \... | Proof. We use the symmetry to extend \( v \) to all of \( \Omega \) . We show that the resulting extension (also called \( v \) ) is continuous on \( \Omega \), harmonic on \( {\Omega }^{ + } \cup {\Omega }^{ - } \), and vanishes on \( \sigma \) .\n\nHarmonicity of \( v \) in \( \Omega \) is a local property; therefore... | Yes |
Every bounded domain \( D \subset \mathbb{C} \) is hyperbolic. | Choose any \( c \notin D \) and \( R > 0 \) so that \( D \) is contained in \( U\left( {c, R}\right) \), and observe that the function \( z \mapsto \) \( \log \left| \frac{z - c}{R}\right| \) is nonconstant, negative, and harmonic (hence subharmonic) on \( D \) . | Yes |
Lemma 9.30. Let \( D \) be a domain in \( \mathbb{C} \) and \( U \) be an open disc such that \( \operatorname{cl}U \subset D \) . If \( u \) is subharmonic in \( D \), then so is \( {u}_{U} \) . | Proof. It suffices to show that \( {u}_{U} \) satisfies the mean value inequality (9.15) at every point \( c \) in \( \partial U \) . Since \( u \) is subharmonic in \( D, u\left( z\right) \leq {u}_{U}\left( z\right) \) for all \( z \) in \( D \) . But then\n\n\[ \n{u}_{U}\left( c\right) = u\left( c\right) \leq \frac{1... | Yes |
Theorem 9.42. Let \( D \) be a nonempty domain in \( \mathbb{C} \). There exists a Green’s function on \( D \) with singularity at some point \( c \in D \) if and only if \( D \) is hyperbolic. In the latter cases \( D \) has a Green’s function with a singularity at any arbitrary point of \( D \). | Proof. Assume that there exists a Green’s function on \( D \) with singularity at some point \( c \in D \). It involves no loss of generality (by Exercise 9.25) to assume that \( D \) contains the unit disc and that \( c = 0 \). Let \( g \) be the Green’s function for \( D \) with singularity at 0. Then \( \mathop{\lim... | Yes |
Theorem 9.44. Let \( D \subset \mathbb{C} \) be a domain with nonempty \( \partial D \) that is regular for the Dirichlet problem. Then \( D \) is hyperbolic.\n\nIn particular, for every \( c \in D \), let \( u \in \mathbf{C}\left( {D \cup \partial D}\right) \) be a harmonic function in \( D \) with \( u\left( z\right)... | Proof. It suffices, of course, to prove the particular claim. Without loss of generality \( \mathbb{D} = \{ z \in \mathbb{C};\left| z\right| < 1\} \subseteq D \) and \( c = 0 \) .\n\nLet \( G\left( z\right) = u\left( z\right) - \log \left| z\right| \) for \( z \in D \cup \partial D - \{ 0\} \) . Observe that the hypoth... | Yes |
Lemma 10.2. Let \( {\left\{ {u}_{n}\right\} }_{n = 1}^{\infty } \subset \mathbb{C} \) and set\n\n\[ \n{p}_{N} = \mathop{\prod }\limits_{{n = 1}}^{N}\left( {1 + {u}_{n}}\right) \text{ and }{p}_{N}^{ * } = \mathop{\prod }\limits_{{n = 1}}^{N}\left( {1 + \left| {u}_{n}\right| }\right) .\n\]\n\nThen\n\n\[ \n{p}_{N}^{ * } \... | Proof. We know that \( x > 0 \) implies that \( {\mathrm{e}}^{x} \geq 1 + x \) . Therefore, \( 1 + \left| {u}_{n}\right| \leq {\mathrm{e}}^{\left| {u}_{n}\right| } \) so that \( {p}_{N}^{ * } \leq {\mathrm{e}}^{\left| {u}_{1}\right| + \cdots + \left| {u}_{N}\right| } \) .\n\nThe second statement is proved by induction ... | Yes |
Theorem 10.4. Assume \( 0 \leq {u}_{n} < 1 \) . (1) If \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n} < \infty \), then \( 0 < \mathop{\prod }\limits_{{n = 1}}^{\infty }\left( {1 - {u}_{n}}\right) < \infty \). (2) If \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n} = + \infty \), then \( \mathop{\prod }\limits_... | Proof. The first claim is a consequence of the previous theorem. To prove the second claim, we start with the observation that \[ 1 - x \leq {\mathrm{e}}^{-x}\text{ for }0 \leq x \leq 1. \] Let \( {p}_{N} = \left( {1 - {u}_{1}}\right) \left( {1 - {u}_{2}}\right) \cdots \left( {1 - {u}_{N}}\right) \). Since \( {p}_{1} \... | Yes |
Theorem 10.5. Let \( D \) be a domain in \( \mathbb{C} \) and suppose that \( \left\{ {f}_{n}\right\} \) is a sequence in \( \mathbf{H}\left( D\right) \) with \( {f}_{n} \) not identically 0 for all \( n \) . (a) If \( \mathop{\sum }\limits_{{n = 1}}^{\infty }\left| {1 - {f}_{n}}\right| \) converges uniformly on compac... | Proof. For part (a), we only have to verify the formula for the order of \( z \) . We note that the sum in that formula is finite (i.e., all but finitely many summands are zero). Let \( {z}_{0} \in D \) and let \( K \subset D \) be a compact set containing a neighborhood of \( {z}_{0} \) . There is an \( N \) in \( {\m... | Yes |
Lemma 10.7. If \( \left| z\right| \leq 1 \), then \( \left| {1 - {E}_{p}\left( z\right) }\right| \leq {\left| z\right| }^{p + 1} \) for all nonnegative integers \( p \) . | Proof. The statement is clearly true if \( p = 0 \) .\n\nIf \( p \geq 1 \), we have\n\n\[ \n{E}_{p}^{\prime }\left( z\right) = \left( {1 - z}\right) {\mathrm{e}}^{z + \frac{{z}^{2}}{2} + \cdots + \frac{{z}^{p}}{p}}\left\lbrack {1 + z + \cdots + {z}^{p - 1}}\right\rbrack - {\mathrm{e}}^{z + \frac{{z}^{2}}{2} + \cdots + ... | Yes |
Theorem 10.8 (Weierstrass Theorem). Assume that \( \left\{ {z}_{n}\right\} \) is a sequence of nonzero complex numbers with \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\left| {z}_{n}\right| = \infty \) . If \( \left\{ {p}_{n}\right\} \subseteq {\mathbb{Z}}_{ \geq 0} \) is a sequence of nonnegative integers with t... | Proof. We first show that (10.1) holds for \( {p}_{n} = n - 1 \) . In this case we have to show convergence of the series \( \sum {a}_{n} \), with \( {a}_{n} = {\left( \frac{r}{\left| {z}_{n}\right| }\right) }^{n} \) . But \( {\left| {a}_{n}\right| }^{\frac{1}{n}} \rightarrow 0 \) as \( n \rightarrow \infty \) , and th... | Yes |
Theorem 10.9 (Weierstrass Factorization Theorem). Let \( f \) be in \( \mathbf{H}\left( \mathbb{C}\right) - \{ 0\} \) , and set \( k = {v}_{0}\left( f\right) \) . Let \( \left\{ {{z}_{n};n \in I}\right\} \) denote the zeros of \( f \) in \( \mathbb{C} - \{ 0\} \), listed according to their multiplicities. There exist a... | Proof. Observe that \( I \subseteq \mathbb{N} \) may be finite (including the possibility that \( I \) is empty) or countable. In the finite case the theorem has already been established. In any case, we can choose any sequence \( \left\{ {{p}_{n};n \in I}\right\} \) of nonnegative integers such that (10.1) holds for a... | Yes |
Theorem 10.16 (Stirling’s Formula). For \( \Re z > 0 \) ,\n\n\[ \Gamma \left( z\right) = \sqrt{2\pi }{z}^{z - \frac{1}{2}}{\mathrm{e}}^{-z}{\mathrm{e}}^{J\left( z\right) }.\n\] | Proof. We know from (10.13) that\n\n\[ \Gamma \left( z\right) = {\mathrm{e}}^{{C}^{\prime } + {Cz}}{z}^{z - \frac{1}{2}}{\mathrm{e}}^{J\left( z\right) },\n\]\n\nand we only need to determine the constants \( {C}^{\prime } \) and \( C \), which we do using (10.15) and the two functional equations (10.8) and (10.9) alrea... | No |
Corollary 10.17.\n\n\\[ \n\\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}\\frac{n!}{\\sqrt{2\\pi n}{\\left( \\frac{n}{e}\\right) }^{n}} = 1 \n\\] | Proof. Note that\n\n\\[ \n{\\left( n + 1\\right) }^{n + \\frac{1}{2}} = {n}^{n + \\frac{1}{2}}{\\left( 1 + \\frac{1}{n}\\right) }^{n}{\\left( 1 + \\frac{1}{n}\\right) }^{\\frac{1}{2}}, \n\\]\n\nand therefore\n\n\\[ \n\\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}\\frac{{\\left( n + 1\\right) }^{n + \\frac{1}{2}}}... | Yes |
Theorem 10.19. Let \( D \) be a nonempty domain in \( \mathbb{C} \) . The map\n\n\[ \n\\left( \\cdot \\right) : \\mathbf{M}\\left( D\\right) - \\{ 0\\} \\rightarrow \\operatorname{Div}\\left( D\\right)\n\]\n\nis a surjective homomorphism, whose kernel is the set of nowhere vanishing holomorphic functions on \( D \) . | Proof. Let \( \\mathcal{D} \\in \\operatorname{Div}\\left( D\\right) \) . Since \( I \) is clearly in the image of the map \( \\left( \\cdot \\right) \), we assume \( \\mathcal{D} \\neq I \) . We can then write \( \\mathcal{D} = {\\mathcal{D}}_{1}{\\mathcal{D}}_{2}^{-1} \), where \( {\\mathcal{D}}_{1} \) and \( {\\math... | No |
Theorem 10.20. Let \( I \neq \mathcal{D} = \mathop{\prod }\limits_{i}{a}_{i}^{{v}_{i}} \) be an integral divisor on the unit disc \( \mathbb{D} \) . The Blaschke product \( {B}_{\mathcal{D}} = \mathop{\prod }\limits_{i}{B}_{{a}_{i}}^{{v}_{i}} \) converges to a nonconstant bounded analytic function on \( \mathbb{D} \) i... | Proof. Without loss of generality, we assume that \( {a}_{i} \neq 0 \) for all \( i \) . It is convenient to write the divisor \( \mathcal{D} = \mathop{\prod }\limits_{j}{a}_{j} \), with each \( {a}_{i} \) appearing in this product \( {v}_{i} \) times. Assume (10.18) translated to \( \mathop{\sum }\limits_{j}\left( {1 ... | Yes |
Theorem 10.21 (Jensen's Formula). Let \( f \) be a holomorphic function in a neighborhood \( V \) of \( \mathbb{D} \), and assume that \( f\left( 0\right) \neq 0 \) . For fixed \( 0 < r < 1 \), let \( \left\{ {{a}_{1},\ldots ,{a}_{k}}\right\} \) denote the zeros of \( f \) in the disc \( U\left( {0, r}\right) \) listed... | Proof. For \( a \in \mathbb{D} \), the modified Blaschke factor\n\n\[ {\widehat{B}}_{a}\left( z\right) = \frac{z - a}{1 - \bar{a}z},\;z \in V \]\n\nhas been encountered many times before. Its properties readily imply that\n\n\[ F\left( z\right) = \frac{f\left( z\right) }{\mathop{\prod }\limits_{{j = 1}}^{k}{\widehat{B}... | Yes |
Corollary 10.22 (Jensen's Inequality). Under the hypothesis of the theorem,\n\n\[ \n\log \left| {f\left( 0\right) }\right| \leq \frac{1}{2\pi }{\int }_{0}^{2\pi }\log \left| {f\left( {r{\mathrm{e}}^{i\theta }}\right) }\right| \mathrm{d}\theta .\n\] | Proof. For each \( j \) , \( \log \left| \frac{r}{{a}_{j}}\right| \geq 0 \) . | No |
Theorem 10.23. Let \( \mathcal{D} = \mathop{\prod }\limits_{i}{a}_{i}^{{v}_{i}} \) be an integral divisor on the unit disc \( \mathbb{D} \). There exists a bounded analytic function \( f \) on \( \mathbb{D} \) with divisor equal to \( \mathcal{D} \) if and only if \( \mathop{\sum }\limits_{i}{v}_{i}\left( {1 - \left| {... | Proof. Let \( f \in \mathbf{H}\left( \mathbb{D}\right) \) be a bounded function with \( \left( f\right) = \mathcal{D} \); choose \( M \in {\mathbb{R}}_{ > 0} \) such that \( \left| f\right| \leq M \).\n\nAssume for the moment that \( {a}_{j} \neq 0 \) for all \( j \). We can certainly find a sequence of numbers \( 0 < ... | Yes |
Proposition 1.1.1. Let \( A \) be an algebra with identity \( e \) and with a norm \( \parallel \cdot \parallel \) under which it is a Banach space. Suppose that the multiplication is continuous in each factor separately. Then there exists a norm \( \parallel \cdot {\parallel }_{0} \) on \( A \) that is equivalent to \... | Proof. For \( x \in A \), let \( {L}_{x} : A \rightarrow A \) be the left translation operator \( y \rightarrow {xy} \) . By the continuity assumption, \( {L}_{x} \) is continuous. Moreover, since \( x = {L}_{x}\left( e\right) \) , the map \( x \rightarrow {L}_{x} \) is an isomorphism of \( A \) into \( \mathcal{B}\lef... | Yes |
Let \( X \) be a locally compact Hausdorff space. We denote by \( {C}^{b}\left( X\right) ,{C}_{0}\left( X\right) \), and \( {C}_{c}\left( X\right) \), respectively, the algebras of all continuous complex-valued functions on \( X \) that are bounded, vanish at infinity, or have compact support. The algebra operations ar... | \[ \parallel f{\parallel }_{\infty } = \mathop{\sup }\limits_{{x \in X}}\left| {f\left( x\right) }\right| \;\left( {f \in {C}_{0}\left( X\right) }\right) ,\] the algebras \( {C}^{b}\left( X\right) \) and \( {C}_{0}\left( X\right) \) are commutative Banach algebras, whereas \( {C}_{c}\left( X\right) \) is complete only ... | Yes |
Example 1.1.4. Let \( a, b \in \mathbb{R} \) such that \( a < b \) and \( n \in \mathbb{N} \), and let \( {C}^{n}\left\lbrack {a, b}\right\rbrack \) be the space of all complex-valued functions on \( \left\lbrack {\mathrm{a},\mathrm{b}}\right\rbrack \) which are \( n \) -times continuously differentiable. With pointwis... | \[ \n\parallel {fg}\parallel = \mathop{\sum }\limits_{{k = 0}}^{n}\frac{1}{k!}{\begin{Vmatrix}{\left( fg\right) }^{\left( k\right) }\end{Vmatrix}}_{\infty } = \mathop{\sum }\limits_{{k = 0}}^{n}\frac{1}{k!}{\begin{Vmatrix}\mathop{\sum }\limits_{{j = 0}}^{k}\left( \begin{array}{l} k \\ j \end{array}\right) {f}^{\left( j... | Yes |
For \( f \in C\left( \mathbb{T}\right) \) and \( n \in \mathbb{Z} \), the \( {n}^{th} \) Fourier coefficient \( {c}_{n}\left( f\right) \) is defined by \[ {c}_{n}\left( f\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }f\left( {e}^{it}\right) {e}^{-{int}}{dt}. \] | Then, for each \( n \in \mathbb{Z} \) , \[ {c}_{n}\left( f\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }\left( {\mathop{\sum }\limits_{{k \in \mathbb{Z}}}{c}_{k}{e}^{ikt}}\right) {e}^{-{int}}{dt} \] \[ = \frac{1}{2\pi }\mathop{\sum }\limits_{{k \in \mathbb{Z}}}{c}_{k}{\int }_{0}^{2\pi }{e}^{i\left( {k - n}\right) t}{dt} ... | Yes |
Lemma 1.1.10. Let \( {\left( {e}_{\lambda }\right) }_{\lambda } \) and \( {\left( {f}_{\mu }\right) }_{\mu } \) be bounded left and right approximate identities for \( A \), respectively. Then the net\n\n\[ \n{\left( {e}_{\lambda } + {f}_{\mu } - {f}_{\mu }{e}_{\lambda }\right) }_{\lambda ,\mu }\n\]\n\nis a bounded app... | Proof. Let \( {g}_{\lambda ,\mu } = {e}_{\lambda } + {f}_{\mu } - {f}_{\mu }{e}_{\lambda } \) . Then, for any \( x \in A \) ,\n\n\[ \n\begin{Vmatrix}{{g}_{\lambda ,\mu }x - x}\end{Vmatrix} = \begin{Vmatrix}{\left( {{e}_{\lambda }x - x}\right) + {f}_{\mu }\left( {x - {e}_{\lambda }x}\right) }\end{Vmatrix} \leq \left( {1... | Yes |
Proposition 1.1.11. Let \( A \) be a normed algebra and let \( M \geq 1 \) . Then the following three conditions are equivalent.\n\n(i) A has left approximate units bounded by \( M \) .\n\n(ii) Given finitely many elements \( {x}_{1},\ldots ,{x}_{n} \) in \( A \) and \( \epsilon > 0 \), there exists \( u \in A \) such ... | Proof. To prove (i) \( \Rightarrow \) (ii), using the formal notation \( \left( {1 - x}\right) y = y - {xy} \) for \( x, y \in A \), we successively choose \( {u}_{1},\ldots ,{u}_{n} \in A \) satisfying \( \begin{Vmatrix}{u}_{j}\end{Vmatrix} \leq M \) and\n\n\[ \begin{Vmatrix}{\left( {1 - {u}_{j}}\right) \cdot \ldots \... | Yes |
Lemma 1.2.5. For every \( x \in A, r\left( x\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}{\begin{Vmatrix}{x}^{n}\end{Vmatrix}}^{1/n} \) . | Proof. It suffices to show that, given \( \epsilon > 0 \), there exists \( N\left( \epsilon \right) \in \mathbb{N} \) such that \( {\begin{Vmatrix}{x}^{n}\end{Vmatrix}}^{1/n} < r\left( x\right) + \epsilon \) for all \( n \geq N\left( \epsilon \right) \) . By definition of \( r\left( x\right) \), there exists \( k \in \... | Yes |
Lemma 1.2.6. Let \( A \) be a Banach algebra with identity \( e \) and let \( x \in A \) with \( r\left( x\right) < 1 \) . Then \( e - x \) is invertible in \( A \) and\n\n\[{\left( e - x\right) }^{-1} = e + \mathop{\sum }\limits_{{n = 1}}^{\infty }{x}^{n}\] | Proof. Fix any \( \eta \) such that \( r\left( x\right) < \eta < 1 \) . By Lemma 1.2.5, \( {\begin{Vmatrix}{x}^{n}\end{Vmatrix}}^{1/n} \leq \eta \) for all \( n \geq N \) for some \( N \in \mathbb{N} \) . Then \( \begin{Vmatrix}{x}^{n}\end{Vmatrix} \leq {\eta }^{n} \) for \( n \geq N \), and since \( \eta < 1 \) , the ... | Yes |
Lemma 1.2.7. Let \( A \) be a normed algebra with identity \( e \). (i) If \( x, y \in G\left( A\right) \) are such that \( \parallel y - x\parallel \leq \frac{1}{2}{\begin{Vmatrix}{x}^{-1}\end{Vmatrix}}^{-1} \), then \[ \begin{Vmatrix}{{y}^{-1} - {x}^{-1}}\end{Vmatrix} \leq 2{\begin{Vmatrix}{x}^{-1}\end{Vmatrix}}^{2}\... | Proof. (i) If \( x \) and \( y \) are as in (i), then \[ \begin{Vmatrix}{y}^{-1}\end{Vmatrix} - \begin{Vmatrix}{x}^{-1}\end{Vmatrix} \leq \begin{Vmatrix}{{y}^{-1} - {x}^{-1}}\end{Vmatrix} = \begin{Vmatrix}{{y}^{-1}\left( {x - y}\right) {x}^{-1}}\end{Vmatrix} \leq \frac{1}{2}\begin{Vmatrix}{y}^{-1}\end{Vmatrix}, \] when... | Yes |
Theorem 1.2.9. Let \( A \) be a Banach algebra with identity \( e \), and suppose that every nonzero element \( x \) of \( A \) is invertible. Then \( A \) is isomorphic to the field of complex numbers. | Proof. By Theorem 1.2.8, for every \( x \in A \), there exists \( {\lambda }_{x} \in \mathbb{C} \) such that \( {\lambda }_{x}e - x \notin G\left( A\right) \) . Since \( G\left( A\right) = A \smallsetminus \{ 0\} \) by hypothesis, it follows that \( {\lambda }_{x}e = x \) . Then, of course, \( {\lambda }_{x} \) is uniq... | Yes |
Lemma 1.2.10. Let \( p \) be a complex polynomial (without constant term if \( A \) does not have an identity). Then, for every element \( x \in A \) , \[ {\sigma }_{A}\left( {p\left( x\right) }\right) = p\left( {{\sigma }_{A}\left( x\right) }\right) = \left\{ {p\left( \lambda \right) : \lambda \in {\sigma }_{A}\left( ... | Proof. Suppose first that \( A \) has an identity \( e \) . If \( p \) is constant, say \( p = \alpha \) , then \( p\left( x\right) = {\alpha e} \) and hence \( {\sigma }_{A}\left( {p\left( x\right) }\right) = {\sigma }_{A}\left( {\alpha e}\right) = \{ \alpha \} = p\left( {{\sigma }_{A}\left( x\right) }\right) \) . So ... | Yes |
Lemma 1.2.11. Let \( A \) be a Banach algebra with identity \( e \) and \( B \) a closed subalgebra of \( A \) containing e. If \( x \in B \), then\n\n\[ \n{\sigma }_{A}\left( x\right) \subseteq {\sigma }_{B}\left( x\right) \text{and}\partial \left( {{\sigma }_{B}\left( x\right) }\right) \subseteq \partial \left( {{\si... | Proof. The first inclusion is immediate from the fact that if \( {\lambda e} - x \in G\left( B\right) \) , then \( {\lambda e} - x \in G\left( A\right) \) . For the second inclusion it suffices to show that \( \partial \left( {{\sigma }_{B}\left( x\right) }\right) \subseteq {\sigma }_{A}\left( x\right) \), because then... | Yes |
Theorem 1.2.12. Let \( A \) be a Banach algebra with identity \( e \) and let \( x \in A \) . Then the following conditions are equivalent.\n\n(i) \( {\rho }_{A}\left( x\right) \) is connected.\n\n(ii) \( {\sigma }_{A}\left( x\right) = {\sigma }_{B}\left( x\right) \) for every closed subalgebra \( B \) of \( A \) conta... | Proof. Suppose that (i) holds and let \( B \) be any subalgebra of \( A \) as in (ii). We have to show that \( {\sigma }_{B}\left( x\right) \subseteq {\sigma }_{A}\left( x\right) \), equivalently, that \( {\sigma }_{B}\left( x\right) \cap {\rho }_{A}\left( x\right) = \varnothing \) . Assume that there exists \( \lambda... | Yes |
Lemma 1.2.13. Let \( A \) be a normed algebra and suppose that \( x, y \in A \) are such that \( {xy} = {yx} \) . Then \( r\left( {xy}\right) \leq r\left( x\right) r\left( y\right) \) and \( r\left( {x + y}\right) \leq r\left( x\right) + r\left( y\right) \) . | Proof. Since \( {\left( xy\right) }^{n} = {x}^{n}{y}^{n} \) for all \( n \in \mathbb{N} \), Lemma 1.2.5 yields that\n\n\[ r\left( {xy}\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}{\begin{Vmatrix}{x}^{n}{y}^{n}\end{Vmatrix}}^{1/n} \leq \mathop{\lim }\limits_{{n \rightarrow \infty }}{\begin{Vmatrix}{x}^{n}\en... | Yes |
Lemma 1.3.3. Let \( C \) be a compact subset of \( G \) . Then there exist positive real numbers \( a \) and \( b \) such that \( a \leq \omega \left( x\right) \leq b \) for all \( x \in C \) . | Proof. We first establish the existence of \( b \) . To that end, for \( n \in \mathbb{N} \), let\n\n\[ \n{U}_{n} = \{ x \in G : \omega \left( x\right) < n\} .\n\]\n\nThen \( \mathop{\bigcup }\limits_{{n = 1}}^{\infty }{U}_{n} = G \) and the sets \( {U}_{n} \) are measurable. Choose \( n \in \mathbb{N} \) such that \( ... | Yes |
Corollary 1.3.4. Let \( \omega \) be a weight function on a compact group \( G \) . Then \( \omega \left( x\right) \geq 1 \) for all \( x \in G \) . | Proof. Assume that \( \omega \left( x\right) < 1 \) for some \( x \in G \) . Since \( \omega \left( {x}^{n}\right) \leq \omega {\left( x\right) }^{n} \) for all \( n \) , we obtain \( \omega \left( {x}^{n}\right) \rightarrow 0 \) as \( n \rightarrow \infty \) . However, \( G \) being compact, \( \omega \) is bounded aw... | Yes |
Lemma 1.3.5. Let \( G \) be a locally compact group and \( \omega \) a weight on \( G \) . (i) Every compactly supported function in \( {L}^{1}\left( G\right) \) belongs to \( {L}^{1}\left( {G,\omega }\right) \) . | Proof. (i) is immediate since \( \omega \) is bounded on compact subsets of \( G \) by Lemma 1.3.3. | Yes |
Lemma 1.3.6. Let \( \omega \) be a weight on \( G \) and \( f \in {L}^{1}\left( {G,\omega }\right) \). (i) For every \( x \in G,{L}_{x}f \in {L}^{1}\left( {G,\omega }\right) \) and \( {\begin{Vmatrix}{L}_{x}f\end{Vmatrix}}_{1,\omega } \leq \omega \left( x\right) \parallel f{\parallel }_{1,\omega } \). (ii) The map \( x... | Proof. (i) follows simply from submultiplicativity of \( \omega \): \[ {\begin{Vmatrix}{L}_{x}f\end{Vmatrix}}_{1,\omega } = {\int }_{G}\left| {f\left( {{x}^{-1}t}\right) }\right| \omega \left( t\right) {dt} \] \[ = {\int }_{G}\left| {f\left( {{x}^{-1}t}\right) }\right| \omega \left( {{x}^{-1}t}\right) \frac{\omega \lef... | Yes |
Lemma 1.4.2. Every proper modular ideal is contained in a maximal modular ideal. | Proof. Let \( I \) be a proper modular ideal and let \( u \in A \) be an identity modulo \( I \) . Let \( \mathcal{L} \) be the set of all ideals \( L \) of \( A \) such that \( I \subseteq L \) and \( u \notin L \) . Then \( \mathcal{L} \) is nonempty since \( I \in \mathcal{L} \) . We order \( \mathcal{L} \) by inclu... | Yes |
Lemma 1.4.4. Let \( A \) be a normed algebra and \( I \) a closed ideal in \( A \) . Then \( A/I \), equipped with the quotient norm, is a normed algebra. If \( A \) is complete, then so is \( A/I \) . | Proof. Because \( A/I \), with the quotient norm, is a normed space and complete whenever \( A \) is, it only remains to observe that the quotient norm is submultiplicative. Now, for \( x, y \in A \) ,\n\n\[ \parallel \left( {x + I}\right) \left( {y + I}\right) \parallel = \parallel {xy} + I\parallel = \mathop{\inf }\l... | Yes |
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