Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
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Proposition 2.1 For a \( * \) -ring \( R \), the following are equivalent:\n\n(1) \( R \) is \( * \) -reflexive.\n\n(2) \( {eRe} \) is \( * \) -reflexive for any projection \( e \in R \) .\n\n(3) \( {eR} \) is \( * \) -reflexive for any central idempotent \( e \in R \) . | Proof \( \left( 1\right) \Rightarrow \left( 2\right) \) : Assume that \( R \) is a \( * \) -reflexive ring and \( {e}^{2} = e = {e}^{ * } \in R \) . Let \( a, b \in \) \( R \) . Then eae, \( {ebe} \in {eRe} \) . If \( 0 = {eae}\left( {eRe}\right) {ebe} = {eaeRebe} \), then we have \( 0 = {\left( ebe\right) }^{ * }{Reae... | No |
Proposition 3.1 Let \( R \) be a \( * \) -reflexive ring and \( a \in R \) be central group invertible. Then \( a \) is Moore-Penrose invertible and \( {a}^{ \dagger } \) is the central group inverse of \( a \) . | Proof This follows directly from Proposition 1.4. | No |
Theorem 1.1 \( f\left( 1\right) = 4 \) and the only configuration that achieves this is a regular tetrahedron. | Proof It is easy to know that we can place at most three points in the plane to determine exactly one distance. Also, the three points are the vertices of an equilateral triangle. According to this, they will create another new distance if we add a point to the plane where the equilateral triangle is located. Thus, we ... | Yes |
Theorem 2.1 \( f\left( 2\right) = 6 \) and there are six optimal 2-distance configurations in \( {\mathbb{E}}^{3} \) . | Proof According to Observation 0.1, we know that the six 2-distance 4-point sets in the plane are formed by some equilateral and isosceles triangles. Therefore, each face of the polytope generated by the optimal 2-distance sets in \( {\mathbb{E}}^{3} \) must be equilateral or isosceles triangles or their combination. O... | Yes |
Theorem 3.1 \( f\left( 3\right) \geq {12} \) . | Proof Case 1: We know that at most 7 points can be placed in the plane to determine exactly three distinct distances. The optimal configurations are regular heptagon and regular hexagon with its center. Suppose that the regular heptagon has three different distances \( a, b, c \) .\n\n![baea7b23-9e02-4820-b749-fd672dd1... | Yes |
Proposition 2.1 The following statements are true.\n\n(1) \( p \in {\Gamma }^{ * } \) is left cycle-free if and only if none of the arrows in \( p \) is eventually landing on or sits inside a cycle in \( {\Gamma }^{ * } \) . | Proof (1) It is obvious if \( p \) is trivial. Otherwise, the claim comes from the observation that an arrow in \( {\Gamma }^{ * } \) is left cycle-isolated if and only if it is neither eventually landing on nor sits inside a cycle in \( {\Gamma }^{ * } \) . | No |
(1) Any left cycle-free path \( p \in {\Gamma }^{ * } \) is contained in some finite dimension subtree of \( \Delta \left( \Lambda \right) \) (with the identification under the labeling maps). | Proof (1) From Proposition 2.1 (2), \( p \) must be contained in some finite dimension subtree \( \Delta \left( \alpha \right) \) of \( \Delta \left( {\Lambda }^{ * }\right) \) . Note that the relation ideal \( {I}^{ * } \) inherits all the relations in \( I \) (identify the arrows in \( {\Gamma }^{ * } \) with the pat... | Yes |
(1) \( \mathrm{D}\left( \Lambda \right) = \mathrm{D}\left( {\Lambda }^{ * }\right) \) . | Proof (1) We show first that \( \mathrm{D}\left( \Lambda \right) = \sup \left\{ {l\left( p\right) \mid p \in {\Gamma }^{ * }}\right\} \), then the claim comes from Proposition 2.2 (3). If there is no \ | No |
Lemma 3.1 The product of each pair of idempotents of \( S \) is equal to 0 . In other words, for any \( {\left( {e}_{\alpha }\right) }_{ii},{\left( {e}_{\beta }\right) }_{jj} \in S \) , | \[ {\left( {e}_{\alpha }\right) }_{ii}{\left( {e}_{\beta }\right) }_{jj} = \left\{ \begin{array}{ll} {\left( {e}_{\alpha }\right) }_{ii}\left( { = {\left( {e}_{\beta }\right) }_{jj}}\right) , & \text{ if }i = j; \\ 0, & \text{ otherwise. } \end{array}\right. \] | Yes |
Lemma 3.2 If \( S \) is semimodular, then for every \( \alpha \in \Gamma ,{M}_{\alpha \alpha } \) is a periodic group. | Proof It is easy to check that \( {\mathcal{M}}_{\alpha } = \left\{ {{\left( a\right) }_{ii} : a \in {M}_{\alpha \alpha }}\right\} \) is a subgroup of \( S \) for a given \( i \in {I}_{\alpha } \) . And, the mapping defined by \( {\left( a\right) }_{ii} \mapsto a \) is an isomorphism of \( {\mathcal{M}}_{\alpha } \) on... | Yes |
Lemma 3.6 Any weak Brandt semigroup of type D is distributive. | Proof Let \( S \) be a weak Brandt semigroup of type D. Then \( S \) is not a Brandt semigroup and of course not a 0 -group, since \( \left| \Gamma \right| \geq 2 \) . Now, denote by \( \Pi \) the set of sinks of \( Q\left( S\right) \) and let \( \Phi = \Gamma \smallsetminus \Pi ,{I}_{\alpha } = \left\{ {i}_{\alpha }\r... | Yes |
Theorem 3.1 Let \( S = \mathcal{M}\left( {{M}_{\alpha \beta };I, I,\Gamma ;P}\right) \) be a weak Brandt semigroup. Then \( S \) is semimodular if and only if \( S \) is one of the following cases:\n\n(i) \( S \smallsetminus \{ 0\} \) is a periodic group whose lattice of subgroups is semimodular;\n\n(ii) \( S \) is iso... | Proof Assume that \( S \) is semimodular. We consider two cases:\n\nIf \( \left| \Gamma \right| \geq 2 \), then by Lemmas 3.4 and \( {3.5}, S \) is type D.\n\nIf \( \left| \Gamma \right| = 1 \), then \( S = \left\{ {{\left( a\right) }_{ij} : a \in {M}_{\alpha \alpha }}\right\} \sqcup \{ 0\} \) . By Lemma 3.2, \( {M}_{\... | Yes |
Corollary 4.1 Let \( S \) be an ample semigroup. If \( S \) is primitively semisimple, then \( S \) is modular (respectively, distributive) if and only if for all \( a \in S \), one of the following three cases hold:\n\n(i) \( {J}^{ * }\left( a\right) /{I}^{ * }\left( a\right) \) is a periodic group whose lattice of su... | Proof Because \( S \) is primitively semisimple, any principal \( * \) -factor of \( S \) is a weak Brandt semigroup. The rest follows from Proposition 1.3 and Corollary 3.1. | No |
Lemma 2.2 Let \( S \) be an ordered \( * \) -semihypergroup with order preserving involution \( * \) . Then\n\n(1) \( (A \circ B{\rbrack }^{ * } = \left( {{B}^{ * } \circ {A}^{ * }}\right\rbrack \) for any \( A, B \subseteq S \) ; | Proof (1) Let \( x \in (A \circ B{\rbrack }^{ * } \) . Then \( {x}^{ * } \leq h \) for some \( h \in A \circ B \) . Since \( * \) is an order preserving involution, \( x \leq {h}^{ * } \in {B}^{ * } \circ {A}^{ * } \) . Thus \( x \in \left( {{B}^{ * } \circ {A}^{ * }}\right\rbrack \) . On the other hand, if \( y \in \l... | Yes |
Theorem 2.1 Let \( S \) be an ordered \( * \) -semihypergroup with order preserving involution \( * \) and \( I \) a hyperideal of \( S \) . Then \( I \) is prime if and only if it is semiprime and weakly prime. In particular, if \( S \) is commutative, then the prime and weakly prime hyperideals concide. | Proof \( \left( \Rightarrow \right) \) It is obtained from Remark 2.1 \( \left( 1\right) \) .\n\n\( \left( \Leftarrow \right) \) Let \( a, b \in S \) and \( a \circ b \subseteq I \) . Then \( (b \circ S \circ a\rbrack \circ (b \circ S \circ a\rbrack \subseteq (b \circ S \circ a \circ b \circ S \circ a\rbrack \subseteq ... | Yes |
Lemma 1.1 The collection \( {\left\{ {\phi }_{j}\right\} }_{j = 1}^{M} \) is an R-dual of \( {\left\{ {\psi }_{i}\right\} }_{i = 1}^{N} \) with respect to \( {\left\{ {e}_{i}\right\} }_{i = 1}^{N} \) and \( {\left\{ {\varepsilon }_{j}\right\} }_{j = 1}^{M} \) if and only if \( {\phi }_{j} = \mathop{\sum }\limits_{{i = ... | Proof Assume \( {\psi }_{i} = \mathop{\sum }\limits_{{j = 1}}^{M}\left\langle {{e}_{i},{\phi }_{j}}\right\rangle {\varepsilon }_{j} \) . Then we have\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{N}\left\langle {{\varepsilon }_{j},{\psi }_{i}}\right\rangle {e}_{i} = \mathop{\sum }\limits_{{i = 1}}^{N}\left\langle {{\varepsilo... | Yes |
Theorem 1.1 If \( {\left\{ {\phi }_{j}\right\} }_{j = 1}^{M} \) is an R-dual of \( {\left\{ {\psi }_{i}\right\} }_{i = 1}^{N} \) with respect to \( {\left\{ {e}_{i}\right\} }_{i = 1}^{N} \) and \( {\left\{ {\varepsilon }_{j}\right\} }_{j = 1}^{M} \) , then \( {\left\{ {\phi }_{j}\right\} }_{j = 1}^{M} \) is a frame for... | Proof Assume that \( {\left\{ {\phi }_{j}\right\} }_{j = 1}^{M} \) is a frame for \( {H}^{N} \), then there exist positive numbers \( A \) and \( B \) such that\n\n\[ A\parallel f{\parallel }^{2} \leq \mathop{\sum }\limits_{{j = 1}}^{M}{\left| \left\langle f,{\phi }_{j}\right\rangle \right| }^{2} \leq B\parallel f{\par... | Yes |
Lemma 2.1 The spanning sets of \( {\left\{ {\psi }_{i}\right\} }_{i = 1}^{N} \) and \( {\left\{ {n}_{j}\right\} }_{j = 1}^{M} \) are equal. | Proof Since \( {n}_{j} = \mathop{\sum }\limits_{{i = 1}}^{N}\left\langle {{\phi }_{j},{e}_{i}}\right\rangle {\widetilde{\psi }}_{i} \), we have\n\n\[\n\operatorname{span}{\left\{ {n}_{j}\right\} }_{j = 1}^{M} \subseteq \operatorname{span}{\left\{ {\widetilde{\psi }}_{i}\right\} }_{i = 1}^{N} = \operatorname{span}{\left... | Yes |
Theorem 2.3 Let \( {\left\{ {\phi }_{j}\right\} }_{j = 1}^{M} \) be a frame for \( {H}^{N},{\left\{ {\psi }_{i}\right\} }_{i = 1}^{N} \) be a Riesz sequence in \( {H}^{M} \) and \( \operatorname{span}{\left\{ {\psi }_{i}\right\} }_{i = 1}^{N} = W \) . Then the followings are equivalent:\n\n(i) \( {\left\{ {\psi }_{i}\r... | Proof (i) \( \Rightarrow \) (ii). Suppose that \( {\left\{ {\psi }_{i}\right\} }_{i = 1}^{N} \) is an R-dual of \( {\left\{ {\phi }_{j}\right\} }_{j = 1}^{M} \) . Then there exist orthonormal basis \( {\left\{ {e}_{i}\right\} }_{i = 1}^{N} \) for \( {H}^{N} \) and orthonormal basis \( {\left\{ {\varepsilon }_{j}\right\... | Yes |
Example 2.4 Let \( M \) be a manifold with a 3 -form \( \omega \in {\Omega }^{3}\left( M\right) \) . Then \( {TM} \oplus {T}^{ * }M \) with the bilinear form and anchor the same as in Example 2.3 and the bracket \( \llbracket \cdot , \cdot {\rrbracket }_{\omega } \) is a Courant algebroid if and only if \( {d}_{\mathrm... | By Theorem 2.3, from a manifold \( M \) with a closed 3 -form \( \omega \in {\Omega }^{3}\left( M\right) \), we get a Lie 2-algebra \( {C}^{\infty }\left( M\right) \xrightarrow[]{{d}_{\mathrm{{dR}}}}\mathfrak{X}\left( M\right) \oplus {\Omega }^{1}\left( M\right) \) . Denote it by \( {L}_{\infty }\left( {{TM} \oplus {T}... | Yes |
Example 5.5 Let \( \mathfrak{g} \) be a Lie algebra which acts on a manifold \( M \) by the Lie algebra homomorphism \( \phi : \mathfrak{g} \rightarrow \mathfrak{X}\left( M\right) \) . Then there is a Lie algebroid structure on the trivial bundle \( A \mathrel{\text{:=}} \mathfrak{g} \times M \rightarrow M \), where th... | \[ {\rho }_{A}\left( {fu}\right) = {f\phi }\left( u\right) \] and the Lie bracket \( {\left\lbrack \cdot , \cdot \right\rbrack }_{A} \) is given by \[ {\left\lbrack fu, gv\right\rbrack }_{A} = {fg}{\left\lbrack u, v\right\rbrack }_{\mathfrak{g}} + {f\phi }\left( u\right) \left( g\right) v - {g\phi }\left( v\right) \lef... | Yes |
Theorem 2.1 The canonical inclusion \( \left( {{\iota }_{1},{\iota }_{2},\cdots }\right) \) of the first persistence complex \( {\Omega }_{ * }^{w}\left( {G}_{ * }\right) \) into the second persistence complex \( {\Gamma }_{ * }^{w}\left( {G}_{ * }\right) \) induces the identity map from the persistence \( R \) - modul... | Proof By Proposition 1.1, the canonical inclusion\n\n\[ \n{\iota }_{n} : {\Omega }_{p}^{{w}_{n}}\left( {G}_{n}\right) \rightarrow {\Gamma }_{p}^{{w}_{n}}\left( {G}_{n}\right) ,\;p \geq 0 \n\]\n\ninduces an isomorphism of the weighted path homology\n\n\[ \n{\left( {\iota }_{n}\right) }_{ * } : {H}_{p}\left( {{G}_{n},{w}... | Yes |
Theorem 3.2 For each \( r \geq 0 \), there is a commutative diagram in which each row is a natural short exact sequence and each sequence splits | Proof By the naturalities of the weighted path homology and the Tor functor, the commutative diagram follows from Theorem 3.1. | No |
Corollary 2.3 Let \( G \) be a finite \( p \) -group. Then\n\n(1) If \( G \) is an outer self dual \( p \) -group, then \( G \) is determined by Theorems 2.1 and 2.2;\n\n(2) If proper sections of \( G \) are all self dual and \( G \) is not self dual, then \( G \) is determined by Corollary 2.2. | Proof (1) Since proper quotient groups of \( G \) are all abelian or isomorphic to \( {M}_{p}\left( {1,1,1}\right) \times \) \( {C}_{p}^{t} \), where \( p \geq 2 \), we have \( \left| {G}^{\prime }\right| = p \) or \( {p}^{2} \) . It follows from the assumption of Theorems 2.1 and 2.2 , we get the conclusion in the sta... | Yes |
Theorem 1.1 Let \( \mathcal{A} \) be a von Neumann algebra without central abelian projections and \( \mathcal{B} \) be a \( * \) -algebra. Let \( \eta \) be a non-zero scalar with \( \eta \neq - 1 \) . Suppose that \( \phi \) is a bijective map from \( \mathcal{A} \) to \( \mathcal{B} \) with \( \phi \left( {{A}_{1}{\... | Proof First we give a key technique. Suppose that \( {A}_{1},{A}_{2},\cdots ,{A}_{m} \) and \( T \) be in \( \mathcal{A} \) with 右适当半群. 显然,右适当半群为 \( {\mathcal{L}}^{ * } \) - 幂单的 (即每个 \( {\mathcal{L}}^{ * } \) 类含唯一幂等元),而左适当半群为 \( {\mathcal{R}}^{ * } \) - 幂单的 (即每个 \( {\mathcal{R}}^{ * } \) 类含唯一幂等元). | No |
Theorem 0.1 Let \( f : G \rightarrow G \) be a graph map with zero topological entropy and let \( \mu \) be a finite and \( f \) -invariant Borel measure on \( G \) . Then for any scrambled set \( S \) of \( f,{\mu }^{ * }\left( S\right) = 0 \) , where \( {\mu }^{ * } \) is the outer measure of \( \mu \) . | It is shown that the same result of Naghmouchi holds for graph maps. Since an interval, a tree and a circle are all graphs, the above result for graphs also holds for them. | No |
Lemma 1.6 For \( i = 1,2,{a}_{l}\left( {H}_{i}\right) \leq 2 \) . | Proof Since \( {H}_{2} \) is a subgraph of \( {H}_{1} \), it suffices to show that \( {a}_{1}\left( {H}_{1}\right) \leq 2 \) . Let \( L \) be a 2-list assignment to the vertices of \( {H}_{1} \) . To construct an \( L \) -forested-coloring \( \phi \) of \( {H}_{1} \), we first color \( {y}_{1} \) with a color \( {a}_{1... | Yes |
Lemma 1.3 For \( f \in {C}_{B}\lbrack 0,\infty ), x \in \lbrack 0,\infty ) \), the following inequality holds:\n\n\[ \n\begin{Vmatrix}{{C}_{n,\alpha }\left( {f, x}\right) }\end{Vmatrix} \leq \parallel f\parallel .\n\] | Proof Since \( {C}_{n,\alpha }\left( {1, x}\right) = 1 \), we get\n\n\[ \n\begin{Vmatrix}{{C}_{n,\alpha }\left( {f, x}\right) }\end{Vmatrix} \leq {C}_{n,\alpha }\left( {1, x}\right) \cdot \parallel f\parallel = \parallel f\parallel .\n\] | Yes |
Lemma 1.4 (i) For \( 0 \leq y < x < {b}_{n} \), there holds\n\n\[{\rho }_{n,\alpha }\left( {\frac{x}{{b}_{n}},\frac{y}{{b}_{n}}}\right) \leq \frac{1}{{\left( x - y\right) }^{2}}{\eta }_{n\alpha }^{2}\left( x\right) .\n\]\n\n\( \left( {1.9}\right) \) | Proof (i) By (1.8), we get\n\n\[{\rho }_{n,\alpha }\left( {\frac{x}{{b}_{n}},\frac{y}{{b}_{n}}}\right) = {\int }_{0}^{y}{\mathrm{\;d}}_{t}{\rho }_{n,\alpha }\left( {\frac{x}{{b}_{n}},\frac{t}{{b}_{n}}}\right)\n\]\n\n\[\leq {\int }_{0}^{y}{\left( \frac{x - t}{x - y}\right) }^{2}{\mathrm{\;d}}_{t}{\rho }_{n,\alpha }\left... | No |
Example 3.1 In this example we consider \( {C}_{3}\left( R\right) \) too. Define\n\n\[ \left( {f}_{0}\right) \Delta = {f}_{0} \otimes {f}_{0} + {f}_{1} \otimes {f}_{2} + {f}_{2} \otimes {f}_{1},\;\left( {f}_{0}\right) \varepsilon = 1; \]\n\n\[ \left( {f}_{1}\right) \Delta = {f}_{0} \otimes {f}_{1} + {f}_{1} \otimes {f}... | Then\n\n\[ \left( {f}_{1}\right) \Delta \left( {\Delta \otimes \mathrm{{id}}}\right) = \left( {{f}_{0} \otimes {f}_{1} + {f}_{1} \otimes {f}_{0} + {f}_{2} \otimes {f}_{2}}\right) \left( {\Delta \otimes \mathrm{{id}}}\right) \]\n\n\[ = \left( {f}_{0}\right) \Delta \otimes {f}_{1} + \left( {f}_{1}\right) \Delta \otimes {... | Yes |
Theorem 3.1 Let \( M = \left\lbrack {Y;{M}_{\alpha },{\theta }_{\alpha ,\beta }}\right\rbrack \) be a strong semilattice of unipotent monoids \( {M}_{\alpha } \) with homomorphism \( {\theta }_{\alpha ,\beta } \) . Let \( I = \mathop{\bigcup }\limits_{{\alpha \in Y}}{I}_{\alpha } \) be a semilattice decomposition of le... | Proof We can observe the fact that the restriction of multiplication on the semigroup \( S = I{\underline{ \times }}_{\varphi }M \) to \( {S}_{\alpha } \) coincides with the multiplication on \( {S}_{\alpha } \) and also \( S \) itself is a semilattice \( Y \) of the semigroups \( {S}_{\alpha } = {I}_{\alpha } \times {... | Yes |
Theorem 1.1. Let \( \alpha ,\beta \) be complex numbers. Then\n\n\[ \overline{\alpha \beta } = \bar{\alpha }\bar{\beta },\;\overline{\alpha + \beta } = \bar{\alpha } + \bar{\beta },\;\overline{\bar{\alpha }} = \alpha . \] | Proof. The proofs follow immediately from the definitions of addition, multiplication, and the complex conjugate. We leave them as exercises (Exercises 3 and 4). | No |
Theorem 1.2. The absolute value of a complex number satisfies the following properties. If \( \alpha ,\beta \) are complex numbers, then\n\n\[ \n\left| {\alpha \beta }\right| = \left| \alpha \right| \left| \beta \right| \n\]\n\n\[ \n\left| {\alpha + \beta }\right| \leqq \left| \alpha \right| + \left| \beta \right| \n\] | Proof. We have\n\n\[ \n{\left| \alpha \beta \right| }^{2} = {\alpha \beta }\overline{\alpha \beta } = \alpha \bar{\alpha }\beta \bar{\beta } = {\left| \alpha \right| }^{2}{\left| \beta \right| }^{2}.\n\]\n\nTaking the square root, we conclude that \( \left| \alpha \right| \left| \beta \right| = \left| {\alpha \beta }\r... | Yes |
Theorem 2.1. Let \( \theta ,\varphi \) be two real numbers. Then\n\n\[ \n{e}^{{i\theta } + {i\varphi }} = {e}^{i\theta }{e}^{i\varphi }\n\] | Proof. By definition, we have\n\n\[ \n{e}^{{i\theta } + {i\varphi }} = {e}^{i\left( {\theta + \varphi }\right) } = \cos \left( {\theta + \varphi }\right) + i\sin \left( {\theta + \varphi }\right) .\n\]\n\nUsing the addition formulas for sine and cosine, we see that the preceding expression is equal to\n\n\[ \n\cos \the... | Yes |
Theorem 2.2. Let \( \alpha ,\beta \) be complex numbers. Then\n\n\[ \n{e}^{\alpha + \beta } = {e}^{\alpha }{e}^{\beta }\n\] | Proof. Let \( \alpha = {a}_{1} + i{a}_{2} \) and \( \beta = {b}_{1} + i{b}_{2} \) . Then\n\n\[ \n{e}^{\alpha + \beta } = {e}^{\left( {{a}_{1} + {b}_{1}}\right) + i\left( {{a}_{2} + {b}_{2}}\right) } = {e}^{{a}_{1} + {b}_{1}}{e}^{i\left( {{a}_{2} + {b}_{2}}\right) }\n\]\n\n\[ \n= {e}^{{a}_{1}}{e}^{{b}_{1}}{e}^{i{a}_{2} ... | Yes |
Find a complex number whose square is \( 4{e}^{{i\pi }/2} \) . | Let \( z = 2{e}^{{i\pi }/4} \) . Using the rule for exponentials, we see that \( {z}^{2} = 4{e}^{{i\pi }/2} \) . | Yes |
Let \( n \) be a positive integer. Find a complex number \( w \) such that \( {w}^{n} = {e}^{{i\pi }/2} \) . | It is clear that the complex number \( w = {e}^{{i\pi }/{2n}} \) satisfies our requirement. | Yes |
Theorem 4.1. A set of complex numbers is compact if and only if it is closed and bounded. | Proof. Assume that \( S \) is compact. If \( S \) is not bounded, for each positive integer \( n \) there exists \( {z}_{n} \in S \) such that\n\n\[ \left| {z}_{n}\right| > n\text{.}\]\n\nThen the sequence \( \left\{ {z}_{n}\right\} \) does not have a point of accumulation. Indeed, if \( v \) is a point of accumulation... | Yes |
Theorem 4.2. Let \( S \) be a compact set and let \( {S}_{1} \supset {S}_{2} \supset \cdots \) be a sequence of non-empty closed subsets such that \( {S}_{n} \supset {S}_{n + 1} \) . Then the intersection of all \( {S}_{n} \) for all \( n = 1,2,\ldots \) is not empty. | Proof. Let \( {z}_{n} \in {S}_{n} \) . The sequence \( \left\{ {z}_{n}\right\} \) has a point of accumulation in \( S \) . Call it \( v \) . Then \( v \) is also a point of accumulation for each subsequence \( \left\{ {z}_{k}\right\} \) with \( k \geqq n \), and hence lies in the closure of \( {S}_{n} \) for each \( n ... | Yes |
Theorem 4.3. Let \( S \) be a compact set of complex numbers, and let \( f \) be a continuous function on \( S \) . Then the image of \( f \) is compact. | Proof. Let \( \left\{ {w}_{n}\right\} \) be a sequence in the image of \( f \), so that\n\n\[ \n{w}_{n} = f\left( {z}_{n}\right) \;\text{ for }\;{z}_{n} \in S.\n\]\n\nThe sequence \( \left\{ {z}_{n}\right\} \) has a convergent subsequence \( \left\{ {z}_{{n}_{k}}\right\} \), with a limit \( v \) in \( S \) . Since \( f... | Yes |
Theorem 4.4. Let \( S \) be a compact set of complex numbers, and let\n\n\[ f : S \rightarrow \mathbf{R} \]\n\nbe a continuous function. Then \( f \) has a maximum on \( S \), that is, there exists \( v \in S \) such that \( f\left( z\right) \leqq f\left( v\right) \) for all \( z \in S \) . | Proof. By Theorem 4.3, we know that \( f\left( S\right) \) is closed and bounded. Let \( b \) be its least upper bound. Then \( b \) is adherent to \( f\left( S\right) \), whence in \( f\left( S\right) \) because \( f\left( S\right) \) is closed. So there is some \( v \in S \) such that \( f\left( v\right) = b \). This... | Yes |
Theorem 4.5. Let \( S \) be a compact set, and let \( f \) be a continuous function on \( S \) . Then \( f \) is uniformly continuous, i.e. given \( \epsilon \) there exists \( \delta \) such that whenever \( z, w \in S \) and \( \left| {z - w}\right| < \delta \), then \( \left| {f\left( z\right) - f\left( w\right) }\r... | Proof. Suppose the assertion of the theorem is false. Then there exists \( \epsilon \), and for each \( n \) there exists a pair of elements \( {z}_{n},{w}_{n} \in S \) such that\n\n\[ \left| {{z}_{n} - {w}_{n}}\right| < 1/n\;\text{ but }\;\left| {f\left( {z}_{n}\right) - f\left( {w}_{n}\right) }\right| > \epsilon . \]... | Yes |
Theorem 4.6. Let \( S \) be a closed set of complex numbers, and let \( v \) be a complex number. There exists a point \( w \in S \) such that\n\n\[ d\left( {S, v}\right) = \left| {w - v}\right| \] | [Hint: Let \( E \) be a closed disc of some suitable radius, centered at \( v \) , and consider the function \( z \mapsto \left| {z - v}\right| \) for \( z \in S \cap E \) .] | No |
Theorem 4.7. Let \( K \) be a compact set of complex numbers, and let \( S \) be a closed set. There exist elements \( {z}_{0} \in K \) and \( {w}_{0} \in S \) such that\n\n\[ d\left( {K, S}\right) = \left| {{z}_{0} - {w}_{0}}\right| \] | [Hint: Consider the function \( z \mapsto d\left( {S, z}\right) \) for \( z \in K \) .] | No |
Theorem 4.8. Let \( S \) be compact. Let \( r \) be a real number \( > 0 \) . There exists a finite number of open discs of radius \( r \) whose union contains \( S \) . | Proof. Suppose this is false. Let \( {z}_{1} \in S \) and let \( {D}_{1} \) be the open disc of radius \( r \) centered at \( {z}_{1} \) . Then \( {D}_{1} \) does not contain \( S \), and there is some \( {z}_{2} \in S,{z}_{2} \neq {z}_{1} \) . Proceeding inductively, suppose we have found open discs \( {D}_{1},\ldots ... | Yes |
Theorem 4.9. Let \( S \) be a compact set, and let \( {\left\{ {U}_{i}\right\} }_{i \in I} \) be an open covering of \( S \) . Then there exists a finite subcovering, that is, a finite number of open sets \( {U}_{{i}_{1}},\ldots ,{U}_{{i}_{n}} \) whose union covers \( S \) . | Proof. By Theorem 4.8, for each \( n \) there exists a finite number of open discs of radius \( 1/n \) which cover \( S \) . Suppose that there is no finite sub-covering of \( S \) by open sets \( {U}_{i} \) . Then for each \( n \) there exists one of the open discs \( {D}_{n} \) from the preceding finite number such t... | Yes |
Theorem 7.1. If \( {f}^{\prime }\left( {z}_{0}\right) \neq 0 \) then the angle between the curves \( \gamma ,\eta \) at \( {z}_{0} \) is the same as the angle between the curves \( f \circ \gamma, f \circ \eta \) at \( f\left( {z}_{0}\right) \) . | Proof. Geometrically speaking, the tangent vectors under \( f \) are changed by multiplication with \( {f}^{\prime }\left( {z}_{0}\right) \), which can be represented in polar coordinates as a dilation and a rotation, so preserves the angles.\n\nWe shall now give a more formal argument, dealing with the cosine and sine... | Yes |
Theorem 2.1. If a sequence \( \left\{ {f}_{n}\right\} \) of functions on \( S \) is Cauchy, then it converges uniformly. | Proof. For each \( z \in S \), let\n\n\[ f\left( z\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}{f}_{n}\left( z\right) \]\n\nGiven \( \epsilon \), there exists \( N \) such that if \( m, n \geqq N \), then\n\n\[ \left| {{f}_{n}\left( z\right) - {f}_{m}\left( z\right) }\right| < \epsilon ,\;\text{ for all }\;... | Yes |
Theorem 2.2. Let \( \\left\\{ {c}_{n}\\right\\} \) be a sequence of real numbers \( \\geqq 0 \), and assume that\n\n\[ \n\\sum {c}_{n}\n\]\n\nconverges. Let \( \\left\\{ {f}_{n}\\right\\} \) be a sequence of functions on \( S \) such that \( \\begin{Vmatrix}{f}_{n}\\end{Vmatrix} \\leqq {c}_{n} \n\nfor all \( n \) . The... | Proof. Say \( m \\leqq n \) . We have an estimate for the difference of partial sums,\n\n\[ \n\\begin{Vmatrix}{{s}_{n} - {s}_{m}}\\end{Vmatrix} \\leqq \\mathop{\\sum }\\limits_{{k = m + 1}}^{n}\\begin{Vmatrix}{f}_{k}\\end{Vmatrix} \\leqq \\mathop{\\sum }\\limits_{{k = m + 1}}^{n}{c}_{k}.\n\]\n\nThe assumption that \( \... | Yes |
Theorem 2.3. Let \( S \) be a set of complex numbers, and let \( \left\{ {f}_{n}\right\} \) be a sequence of continuous functions on \( S \) . If this sequence converges uniformly, then the limit function \( f \) is also continuous. | Proof. You should already have seen this theorem some time during a calculus course. We reproduce the proof for convenience. Let \( \alpha \in S \) . Select \( n \) so large that \( \begin{Vmatrix}{f - {f}_{n}}\end{Vmatrix} < \epsilon \) . For this choice of \( n \), using the continuity of \( {f}_{n} \) at \( \alpha \... | Yes |
Theorem 2.4. Let \( \\left\\{ {a}_{n}\\right\\} \) be a sequence of complex numbers, and let \( r \) be a number \( > 0 \) such that the series\n\n\[ \n\\sum \\left| {a}_{n}\\right| {r}^{n} \n\]\n\nconverges. Then the series \( \\sum {a}_{n}{z}^{n} \) converges absolutely and uniformly for \( \\left| z\\right| \\leqq r... | Proof. Special case of the comparison test. | No |
Theorem 2.5. Let \( \sum {a}_{n}{z}^{n} \) be a power series. If it does not converge absolutely for all \( z \), then there exists a number \( r \) such that the series converges absolutely for \( \left| z\right| < r \) and does not converge absolutely for \( \left| z\right| > r \) . | Proof. Suppose that the series does not converge absolutely for all \( z \) . Let \( r \) be the least upper bound of those numbers \( s \geqq 0 \) such that\n\n\[ \sum \left| {a}_{n}\right| {s}^{n} \]\n\nconverges. Then \( \sum \left| {a}_{n}\right| {\left| z\right| }^{n} \) diverges if \( \left| z\right| > r \), and ... | Yes |
Theorem 2.6. Let \( \sum {a}_{n}{z}^{n} \) be a power series, and let \( r \) be its radius of convergence. Then\n\n\[ \frac{1}{r} = \lim \sup {\left| {a}_{n}\right| }^{1/n} \]\n\nIf \( r = 0 \), this relation is to be interpreted as meaning that the sequence \( \left\{ {\left| {a}_{n}\right| }^{1/n}\right\} \) is not ... | Proof. Let \( t = \limsup {\left| {a}_{n}\right| }^{1/n} \) . Suppose first that \( t \neq 0,\infty \) . Given \( \epsilon > 0 \) , there exist only a finite number of \( n \) such that \( {\left| {a}_{n}\right| }^{1/n} \geqq t + \epsilon \) . Thus for all but a finite number of \( n \), we have\n\n\[ \left| {a}_{n}\ri... | No |
Corollary 2.7. If \( \lim {\left| {a}_{n}\right| }^{1/n} = t \) exists, then \( r = 1/t \) . | Proof. If the limit exists, then \( t \) is the only point of accumulation of the sequence \( {\left| {a}_{n}\right| }^{1/n} \), and the theorem states that \( t = 1/r \) . | No |
Corollary 2.8. Suppose that \( \sum {a}_{n}{z}^{n} \) has a radius of convergence greater than 0 . Then there exists a positive number \( C \) such that if \( A > 1/r \) then \[ \left| {a}_{n}\right| \leqq C{A}^{n}\;\text{ for all }n. \] | Proof. Let \( s = 1/A \) so \( 0 < s < r \) at the beginning of the proof of the theorem. | No |
Theorem 3.1. If \( f, g \) are power series which converge absolutely on the disc \( D\left( {0, r}\right) \), then \( f + g \) and \( {fg} \) also converge absolutely on this disc. If \( \alpha \) is a complex number, \( \alpha \) fonverges absolutely on this disc, and we have\n\n\[ \left( {f + g}\right) \left( z\righ... | Proof. We give the proof for the product, which is the hardest. Let \n\n\[ f = \sum {a}_{n}{T}^{n}\;\text{ and }\;g = {b}_{n}{T}^{n}, \] \n\nso that \n\n\[ {fg} = \sum {c}_{n}{T}^{n},\;\text{ where }\;{c}_{n} = \mathop{\sum }\limits_{{k = 0}}^{n}{a}_{k}{b}_{n - k}. \] \n\nLet \( 0 < s < r \) . We know that there exists... | Yes |
(a) Let \( f\left( T\right) = \sum {a}_{n}{T}^{n} \) be a non-constant power series, having a nonzero radius of convergence. If \( f\left( 0\right) = 0 \), then there exists \( s > 0 \) such that \( f\left( z\right) \neq 0 \) for all \( z \) with \( \left| z\right| \leqq s \), and \( z \neq 0 \) . | Proof. We can write\n\n\[ f\left( z\right) = {a}_{m}{z}^{m} + \text{ higher terms,}\;\text{ and }{a}_{m} \neq 0 \]\n\n\[ = {a}_{m}{z}^{m}\left( {1 + {b}_{1}z + {b}_{2}{z}^{2} + \cdots }\right) \]\n\n\[ = {a}_{m}{z}^{m}\left( {1 + h\left( z\right) }\right) \]\n\nwhere \( h\left( z\right) = {b}_{1}z + {b}_{2}{z}^{2} + \c... | Yes |
Theorem 3.3. Suppose that \( f \) has a non-zero radius of convergence, and non-zero constant term. Let \( g \) be the formal power series which is inverse to \( f \), that is, \( {fg} = 1 \) . Then \( g \) also has a non-zero radius of convergence. | Proof. Multiplying \( f \) by some constant, we may assume without loss of generality that the constant term is 1 , so we write\n\n\[ f = 1 + {a}_{1}T + {a}_{2}{T}^{2} + \cdots = 1 - h\left( T\right) ,\]\n\nwhere \( h\left( T\right) \) has constant term equal to 0 . By Corollary 2.8, we know that there exists a number ... | Yes |
Theorem 3.4. Let\n\n\\[ \nf\left( z\right) = \mathop{\sum }\limits_{{n \geqq 0}}{a}_{n}{z}^{n}\\;\\text{ and }\\;h\left( z\right) = \mathop{\sum }\limits_{{n \geqq 1}}{b}_{n}{z}^{n} \n\\]\n\nbe convergent power series, and assume that the constant term of \\( h \\) is 0 . Assume that \\( f\left( z\right) \\) is absolut... | Proof. Let \\( g\left( T\\right) = \\sum {c}_{n}{T}^{n} \\) . Then \\( g\left( T\\right) \\) is dominated by the series\n\n\\[ \ng\left( T\\right) \\prec \mathop{\sum }\limits_{{n = 0}}^{\infty }\\left| {a}_{n}\\right| {\\left( \mathop{\sum }\limits_{{k = 1}}^{\infty }\\left| {b}_{k}\\right| {T}^{k}\\right) }^{n}\n\\]\... | Yes |
Theorem 4.1. Let \( f\left( z\right) = \sum {a}_{n}{z}^{n} \) be a power series whose radius of convergence is \( r \) . Then \( f \) is analytic on the open disc \( D\left( {0, r}\right) \) . | Proof. We have to show that \( f \) has a power series expansion at an arbitrary point \( {z}_{0} \) of the disc, so \( \left| {z}_{0}\right| < r \) . Let \( s > 0 \) be such that\n\n\( \left| {z}_{0}\right| + s < r \) . We shall see that \( f \) can be represented by a convergent power series at \( {z}_{0} \), converg... | Yes |
Theorem 5.1. If \( f\left( z\right) = \sum {a}_{n}{z}^{n} \) has radius of convergence \( r \), then:\n\n(i) The series \( \sum n{a}_{n}{z}^{n - 1} \) has the same radius of convergence.\n\n(ii) The function \( f \) is holomorphic on \( D\left( {0, r}\right) \), and its derivative is equal to \( \sum n{a}_{n}{z}^{n - 1... | Proof. By Theorem 2.6, we have\n\n\[ \lim \sup {\left| {a}_{n}\right| }^{1/n} = 1/r \]\n\nBut\n\n\[ \lim \sup {\left| n{a}_{n}\right| }^{1/n} = \lim \sup {n}^{1/n}{\left| {a}_{n}\right| }^{1/n}. \]\n\nSince \( \lim {n}^{1/n} = 1 \), the sequences\n\n\[ {\left| n{a}_{n}\right| }^{1/n}\;\text{ and }\;{\left| {a}_{n}\righ... | Yes |
Theorem 6.2. Let \( f \) be analytic on an open set \( U \), and assume that for each point of \( U, f \) is not constant on a given neighborhood of that point. Then \( f \) is an open mapping. | Proof. We apply the preceding discussion to the power series expansion of \( f \) at a point of \( U \), so the proof is obvious in the light of what we have already done. | No |
Theorem 6.4. Let \( f \) be analytic on an open set \( U \), and assume that \( f \) is injective. Let \( V = f\left( U\right) \) be its image. Then \( f : U \rightarrow V \) is an analytic isomorphism, and \( {f}^{\prime }\left( z\right) \neq 0 \) for all \( z \in U \) . | Proof. The function \( f \) between \( U \) and \( V \) is bijective, so we can define an inverse mapping \( g : V \rightarrow U \) . Let \( {z}_{0} \) be a point of \( U \), and let the power series expansion of \( f \) at \( {z}_{0} \) be as in Theorem 6.3. If \( m > 1 \) then we see that \( f \) cannot be injective,... | Yes |
Example 1. Let \( f\left( z\right) = 3 - {5z} + \) higher terms. Then \( f\left( 0\right) = 3 \) | \[ {f}^{\prime }\left( 0\right) = {a}_{1} = - 5 \neq 0. \] Hence \( f \) is a local analytic isomorphism, or locally invertible, at 0 . | Yes |
Let \( f\left( z\right) = 2 - {2z} + {z}^{2} \) . We want to determine whether \( f \) is locally invertible at \( z = 1 \) . | We write the power series expansion of \( f \) at 1 , namely\n\n\[ f\left( z\right) = 1 + {\left( z - 1\right) }^{2} = 1 + {a}_{2}{\left( z - 1\right) }^{2}. \]\n\nHere we have \( {a}_{1} = 0 \) . Hence \( f \) is not locally invertible at \( z = 1 \) . | Yes |
Example 3. Let \( f\left( z\right) = \cos z \) . Determine whether \( f \) is locally invertible at \( z = 0 \) . | In this case, \[ f\left( z\right) = 1 - \frac{{z}^{2}}{2} + \text{ higher terms,} \] so \( {a}_{1} = 0 \) and \( f \) is not locally invertible. | Yes |
Example 4. Let \( f\left( z\right) = {z}^{3} \) . Then \( {f}^{\prime }\left( z\right) = 3{z}^{2} \) and \( {f}^{\prime }\left( 0\right) = 0 \) . Thus \( f \) is not locally invertible at 0 . On the other hand, \( {f}^{\prime }\left( z\right) \neq 0 \) if \( z \neq 0 \) . Hence if \( {z}_{0} \neq 0 \) then \( f \) is l... | ## II, §6. EXERCISES\n\nDetermine which of the following functions are local analytic isomorphism at the given point. Give the reason for your answer.\n\n1. \( f\left( z\right) = {e}^{z} \) at \( z = 0 \) .\n\n2. \( f\left( z\right) = \sin \left( {z}^{2}\right) \) at \( z = 0 \) .\n\n3. \( f\left( z\right) = \left( {z ... | No |
Theorem 7.1. Let \( f \) be analytic on an open set \( U \) . Let \( {z}_{0} \in U \) be a maximum for \( \left| f\right| \), that is,\n\n\[ \left| {f\left( {z}_{0}\right) }\right| \geqq \left| {f\left( z\right) }\right| ,\;\text{ for all }\;z \in U.\n\]\n\nThen \( f \) is locally constant at \( {z}_{0} \) . | Proof. The function \( f \) has a power series expansion at \( {z}_{0} \),\n\n\[ f\left( z\right) = {a}_{0} + {a}_{1}\left( {z - {z}_{0}}\right) + \cdots .\n\]\n\nIf \( f \) is not the constant \( {a}_{0} = f\left( {z}_{0}\right) \), then by Theorem 6.2 we know that \( f \) is an open mapping in a neighborhood of \( {z... | Yes |
Corollary 7.2. Let \( f \) be analytic on an open set \( U \), and let \( {z}_{0} \in U \) be a maximum for the real part \( \operatorname{Re}f \), that is,\n\n\[ \operatorname{Re}f\left( {z}_{0}\right) \geqq \operatorname{Re}f\left( z\right) ,\;\text{ for all }\;z \in U.\n\]\n\nThen \( f \) is locally constant at \( {... | Proof. The function \( {e}^{f\left( z\right) } \) is analytic on \( U \), and if\n\n\[ f\left( z\right) = u\left( z\right) + {iv}\left( z\right) \]\n\n\nis the expression of \( f \) in terms of its real and imaginary parts, then\n\n\[ \left| {e}^{f\left( z\right) }\right| = {e}^{u\left( z\right) }\n\]\n\nHence a maximu... | Yes |
Theorem 7.3. Let\n\n\\[ \nf\\left( z\\right) = {a}_{0} + {a}_{1}z + \\cdots + {a}_{d}{z}^{d} \n\\]\n\nbe a polynomial, not constant, and say \\( {a}_{d} \\neq 0 \\) . Then \\( f \\) has some complex zero, i.e. a number \\( {z}_{0} \\) such that \\( f\\left( {z}_{0}\\right) = 0 \\) . | Proof. Suppose otherwise, so that \\( 1/f\\left( z\\right) \\) is defined for all \\( z \\), and defines an analytic function. Writing\n\n\\[ \nf\\left( z\\right) = {a}_{d}{z}^{d}\\left( {\\frac{{a}_{0}}{{a}_{d}{z}^{d}} + \\frac{{a}_{1}z}{{a}_{d}{z}^{d}} + \\cdots + 1}\\right) , \n\\]\n\none sees that\n\n\\[ \n\\mathop... | Yes |
Theorem 1.1. Let \( U \) be a connected open set, and let \( f \) be a holomorphic function on \( U \) . If \( {f}^{\prime } = 0 \) then \( f \) is constant. | Proof. Let \( \alpha ,\beta \) be two points in \( U \), and suppose first that \( \gamma \) is a curve joining \( \alpha \) to \( \beta \), so that\n\n\[ \n\gamma \left( a\right) = \alpha \;\text{ and }\;\gamma \left( b\right) = \beta .\n\]\n\nThe function\n\n\[ \nt \mapsto f\left( {\gamma \left( t\right) }\right)\n\]... | Yes |
Theorem 1.2. Let \( U \) be a connected open set.\n\n(i) If \( f \) is analytic on \( U \) and not constant, then the set of zeros of \( f \) on \( U \) is discrete.\n\n(ii) Let \( f, g \) be analytic on \( U \) . Let \( S \) be a set of points in \( U \) which is not discrete (so some point of \( S \) is not isolated)... | Proof. We observe that (ii) follows from (i). It suffices to consider the difference \( f - g \) . Therefore we set about to prove (i). We know from Theorem 3.2 of the preceding chapter that either \( f \) is locally constant and equal to 0 in the neighborhood of a zero \( {z}_{0} \), or \( {z}_{0} \) is an isolated ze... | Yes |
Theorem 1.3. Let \( U \) be a connected open set, and let \( f \) be an analytic function on \( U \) . If \( {z}_{0} \in U \) is a maximum point for \( \left| f\right| \), that is\n\n\[ \left| {f\left( {z}_{0}\right) }\right| \geqq \left| {f\left( z\right) }\right| \]\n\nfor all \( z \in U \), then \( f \) is constant ... | Proof. By Theorem 6.1 of the preceding chapter, we know that \( f \) is locally constant at \( {z}_{0} \) . Therefore \( f \) is constant on \( U \) by Theorem 1.2(ii) (compare the constant function and \( f \) ). This concludes the proof. | Yes |
Corollary 1.4. Let \( U \) be a connected open set and \( {U}^{\mathrm{c}} \) its closure. Let \( f \) be a continuous function on \( {U}^{\mathrm{c}} \), analytic and non-constant on \( U \) . If \( {z}_{0} \) is a maximum for \( f \) on \( {U}^{\mathrm{c}} \), that is, \( \left| {f\left( {z}_{0}\right) }\right| \geqq... | Proof. This comes from a direct application of Theorem 1.3. | No |
Lemma 1.5. Let \( S \) be a subset of an open set \( U \) . Then \( S \) is closed in \( U \) if and only if the complement of \( S \) in \( U \) is open, that is, \( U - S \) is open. In particular, if \( S \) is both open and closed in \( U \), then \( U - S \) is also open and closed in \( U \) . | Proof. Exercise 1. | No |
Let \( f\left( z\right) = 1/z \) . Let \( \gamma \left( \theta \right) = {e}^{i\theta } \) . Then we want to find the value of the integral of \( f \) over the circle, \[ {\int }_{\gamma }\frac{1}{z}{dz} \] so \( 0 \leqq \theta \leqq {2\pi } \). | By definition, this integral is equal to \[ {\int }_{0}^{2\pi }\frac{1}{{e}^{i\theta }}i{e}^{i\theta }{d\theta } = i{\int }_{0}^{2\pi }{d\theta } = {2\pi i} \] | Yes |
Theorem 2.1. Let \( f \) be continuous on an open set \( U \), and suppose that \( f \) has a primitive \( g \), that is, \( g \) is holomorphic and \( {g}^{\prime } = f \) . Let \( \alpha ,\beta \) be two points of \( U \), and let \( \gamma \) be a path in \( U \) joining \( \alpha \) to \( \beta \) . Then\n\n\[{\int... | Proof. Assume first that the path is a curve. Then\n\n\[{\int }_{\gamma }f\left( z\right) {dz} = {\int }_{a}^{b}{g}^{\prime }\left( {\gamma \left( t\right) }\right) {\gamma }^{\prime }\left( t\right) {dt}\]\n\nBy the chain rule, the expression under the integral sign is the derivative\n\n\[\frac{d}{dt}g\left( {\gamma \... | Yes |
Example 2. Let \( f\left( z\right) = {z}^{3} \) . Then \( f \) has a primitive, \( g\left( z\right) = {z}^{4}/4 \) . Hence the integral of \( f \) from \( 2 + {3i} \) to \( 1 - i \) over any path is equal to | \[ \frac{{\left( 1 - i\right) }^{4}}{4} - \frac{{\left( 2 + 3i\right) }^{4}}{4} \] | Yes |
Let \( f\left( z\right) = {e}^{z} \) . Find the integral of \( f \) from 1 to in taken over a line segment. | Here again \( {f}^{\prime }\left( z\right) = f\left( z\right) \), so \( f \) has a primitive. Thus the integral is independent of the path and equal to \( {e}^{i\pi } - {e}^{1} = - 1 - e \) . | Yes |
Let \( f\left( z\right) = {z}^{n} \), where \( n \) is an integer \( \neq - 1 \) . Then for any closed path \( \gamma \) (or any closed path not passing through the origin if \( n \) is negative), we have\n\n\[{\int }_{\gamma }{z}^{n}{dz} = 0\] | This is true because \( {z}^{n} \) has the primitive \( {z}^{n + 1}/\left( {n + 1}\right) \) . [When \( n \) is negative, we have to assume that the closed path does not pass through the origin, because the function is then not defined at the origin.] | Yes |
Theorem 2.2. Let \( U \) be a connected open set, and let \( f \) be a continuous function on \( U \) . If the integral of \( f \) along any closed path in \( U \) is equal to 0, then \( f \) has a primitive \( g \) on \( U \), that is, a function \( g \) which is holomorphic such that \( {g}^{\prime } = f \) . | Proof. Pick a point \( {z}_{0} \) in \( U \) and define\n\n\[ g\left( z\right) = {\int }_{{z}_{0}}^{z}f \]\n\nwhere the integral is taken along any path from \( {z}_{0} \) to \( z \) in \( U \) . If \( \gamma ,\eta \) are two such paths, and \( {\eta }^{ - } \) is the reverse path of \( \eta \) (cf. Exercise 9), then \... | Yes |
Theorem 2.3. Let \( f \) be a continuous function on \( U \) . Let \( \gamma \) be a path in U. Then\n\n\[ \left| {{\int }_{\gamma }f}\right| \leqq \parallel f{\parallel }_{\gamma }L\left( \gamma \right) \] | Proof. If \( \gamma \) is a curve, then\n\n\[ \left| {{\int }_{\gamma }f}\right| = \left| {{\int }_{a}^{b}f\left( {\gamma \left( t\right) }\right) {\gamma }^{\prime }\left( t\right) {dt}}\right| \]\n\n\[ \leqq {\int }_{a}^{b}\left| {f\left( {\gamma \left( t\right) }\right) }\right| \left| {{\gamma }^{\prime }\left( t\r... | Yes |
Theorem 2.4. Let \( \\left\\{ {f}_{n}\\right\\} \) be a sequence of continuous functions on \( U \) , converging uniformly to a function \( f \) . Then\n\n\[ \n\\lim {\\int }_{\\gamma }{f}_{n} = {\\int }_{\\gamma }f \n\] | Proof. The first assertion is immediate from the inequality.\n\n\[ \n\\left| {{\\int }_{\\gamma }{f}_{n} - {\\int }_{\\gamma }f}\\right| \\leqq {\\int }_{\\gamma }\\left| {{f}_{n} - f}\\right| \\leqq \\begin{Vmatrix}{{f}_{n} - f}\\end{Vmatrix}L\\left( \\gamma \\right) .\n\] | Yes |
Example 5. Let \( f \) be analytic on an open set containing the closed disc \( \bar{D}\left( {0, R}\right) \) of radius \( R \) centered at the origin, except possibly at the origin. Suppose \( f \) has a power series expansion\n\n\[ f\left( z\right) = \frac{{a}_{-m}}{{z}^{m}} + \cdots + \frac{{a}_{-1}}{z} + {a}_{1}z ... | This is a special case of Theorem 2.4 and Example 4, by letting\n\n\[ {f}_{n}\left( z\right) = \mathop{\sum }\limits_{{k = - m}}^{n}{a}_{k}{z}^{k} \]\n\nEach \( {f}_{n} \) is a finite sum, so the integral of \( {f}_{n} \) is the sum of the integrals of the individual terms, which were evaluated in Example 4. | Yes |
Theorem 3.2. Let \( U \) be a disc centered at a point \( {z}_{0} \). Let \( f \) be continuous on \( U \), and assume that for each rectangle \( R \) contained in \( U \) we have\n\n\[{\int }_{\partial \mathbf{R}}f = 0\]\n\nFor each point \( {z}_{1} \) in the disc, define\n\n\[g\left( {z}_{1}\right) = {\int }_{{z}_{0}... | Proof. We have\n\n\[g\left( {{z}_{1} + h}\right) - g\left( {z}_{1}\right) = {\int }_{{z}_{1}}^{{z}_{1} + h}f\left( z\right) {dz}\]\n\n\n\nFigure 11\n\nThe integral between \( {z}_{1} \) and \( {z}_{1} + h \) is taken... | Yes |
Lemma 4.1. Let \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow U \) be a continuous curve in an open set \( U \) . Then there is some positive number \( r > 0 \) such that every point on the curve lies at distance \( \geqq r \) from the complement of \( U \) . | Proof. The image of \( \gamma \) is compact. Consider the function\n\n\[ \varphi \left( t\right) = \mathop{\min }\limits_{w}\left| {\gamma \left( t\right) - w}\right| \]\n\nwhere the minimum is taken for all \( w \) in the complement of \( U \) . This minimum exists because it suffices to consider \( w \) lying inside ... | Yes |
Lemma 4.2. Let \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow U \) be a continuous curve. Let\n\n\[ \n{a}_{0} = a \leqq {a}_{1} \leqq {a}_{2} \leqq \cdots \leqq {a}_{n} = b \n\]\n\nbe a partition of \( \left\lbrack {a, b}\right\rbrack \) such that the image \( \gamma \left( \left\lbrack {{a}_{i},{a}_{i + 1}}\... | Proof. First let us work with the given partition, but let \( {B}_{i} \) be another disc containing the image \( \gamma \left( \left\lbrack {{a}_{i},{a}_{i + 1}}\right\rbrack \right) \), and \( {B}_{i} \) contained in \( U \) . Let \( {h}_{i} \) be a primitive of \( f \) on \( {B}_{i} \) . Then both \( {g}_{i},{h}_{i} ... | Yes |
Lemma 4.3. Let \( \gamma ,\eta \) be two continuous paths in an open set \( U \), and assume that they have the same beginning point and the same end point. Assume also that they are close together. Let \( f \) be holomorphic on \( U \) . Then\n\n\[ \n{\int }_{\gamma }f = {\int }_{\eta }f \n\] | Proof. We suppose that the paths are defined on the same interval \( \left\lbrack {a, b}\right\rbrack \), and we choose a partition and discs \( {D}_{i} \) as above. Let \( {g}_{i} \) be a primitive of \( f \) on \( {D}_{i} \) . Let\n\n\[ \n{z}_{i} = \gamma \left( {a}_{i}\right) \;\text{ and }\;{w}_{i} = \eta \left( {a... | Yes |
Lemma 4.4. Let \( \gamma ,\eta \) be closed continuous paths in the open set \( U \), say defined on the same interval \( \left\lbrack {a, b}\right\rbrack \) . Assume that they are close together. Let \( f \) be holomorphic on \( U \) . Then\n\n\[ \n{\int }_{\gamma }f = {\int }_{\eta }f \n\] | Proof. The proof is the same as above, except that the reason why we find 0 in the last step is now slightly different. Since the paths are closed, we have\n\n\[ \n{z}_{0} = {z}_{n}\;\text{ and }\;{w}_{0} = {w}_{n}, \n\]\n\nas illustrated in Fig. 15. The two primitives \( {g}_{n - 1} \) and \( {g}_{0} \) differ by a co... | No |
Let \( \gamma ,\eta \) be closed paths in \( U \), and assume that they are homotopic in \( U \). Let \( f \) be holomorphic on \( U \). Then \[ {\int }_{\gamma }f = {\int }_{\eta }f \] In particular, if \( \gamma \) is homotopic to a point in \( U \), then \[ {\int }_{\gamma }f = 0 \] | The formal proof runs as follows. Let \[ \psi : \left\lbrack {a, b}\right\rbrack \times \left\lbrack {c, d}\right\rbrack \rightarrow U \] be the homotopy. The image of \( \psi \) is compact, and hence has distance \( > 0 \) from the complement of \( U \). By uniform continuity we can therefore find partitions \[ a = {a... | Yes |
Lemma 5.3. Let \( S \) be a convex set, and let \( \gamma ,\eta \) be continuous closed curves in \( S \) . Then \( \gamma ,\eta \) are homotopic in \( S \) . | Proof. We define\n\n\[ \psi \left( {t, s}\right) = {s\gamma }\left( t\right) + \left( {1 - s}\right) \eta \left( t\right) \]\n\nIt is immediately verified that each curve \( {\psi }_{s} \) defined by \( {\psi }_{s}\left( t\right) = \psi \left( {t, s}\right) \) is a closed curve, and \( \psi \) is continuous. Also\n\n\[... | Yes |
Theorem 6.1. Let \( f \) be holomorphic on a simply connected open set \( U \) . Let \( {z}_{0} \in U \) . For any point \( z \in U \) the integral\n\n\[ g\left( z\right) = {\int }_{{z}_{0}}^{z}f\left( \zeta \right) {d\zeta } \]\n\nis independent of the path in \( U \) from \( {z}_{0} \) to \( z \), and \( g \) is a pr... | Proof. Let \( {\gamma }_{1},{\gamma }_{2} \) be two paths in \( U \) from \( {z}_{0} \) to \( z \) . Let \( {\gamma }_{2}^{ - } \) be the reverse path of \( {\gamma }_{2} \), from \( z \) to \( {z}_{0} \) . Then\n\n\[ \gamma = \left\{ {{\gamma }_{1},{\gamma }_{2}^{ - }}\right\} \]\n\nis a closed path, and by the first ... | Yes |
Theorem 7.1 (Local Cauchy Formula). Let \( \\bar{D} \) be a closed disc of positive radius, and let \( f \) be holomorphic on \( \\bar{D} \) (that is, on an open disc \( U \) containing \( \\bar{D} \) ). Let \( \\gamma \) be the circle which is the boundary of \( \\bar{D} \) . Then for every \( {z}_{0} \\in D \) we hav... | Proof. Let \( {C}_{r} \) be the circle of radius \( r \) centered at \( {z}_{0} \), as illustrated on Fig. 18.\n\n\n\nFigure 18\n\nThen for small \( r,\\gamma \) and \( {C}_{r} \) are homotopic. The idea for construc... | Yes |
Theorem 7.2. Let \( f \) be holomorphic on an open set \( U \) . Then \( f \) is analytic on \( U \) . | Proof. We must show that \( f \) has a power series expansion at every point \( {z}_{0} \) of \( U \) . Because \( U \) is open, for each \( {z}_{0} \in U \) there is some \( R > 0 \) such that the closed disc \( \bar{D}\left( {{z}_{0}, R}\right) \) centered at \( {z}_{0} \) and of radius \( R \) is contained in \( U \... | Yes |
Theorem 7.3. Let \( f \) be holomorphic on a closed disc \( \bar{D}\left( {{z}_{0}, R}\right), R > 0 \) . Let \( {C}_{R} \) be the circle bounding the disc. Then \( f \) has a power series expansion\n\n\[ f\left( z\right) = \sum {a}_{n}{\left( z - {z}_{0}\right) }^{n} \]\n\nwhose coefficients \( {a}_{n} \) are given by... | Proof. By Theorem 7.1, for all \( z \) inside the circle \( {C}_{R} \), we have\n\n\[ f\left( z\right) = \frac{1}{2\pi i}{\int }_{{c}_{R}}\frac{f\left( \zeta \right) }{\zeta - z}{d\zeta } \]\n\nLet \( 0 < s < R \) . Let \( D\left( {{z}_{0}, s}\right) \) be the disc of radius \( s \) centered at \( {z}_{0} \) . We shall... | Yes |
Corollary 7.4. Let \( f \) be an entire function, and let \( \parallel f{\parallel }_{R} \) be its sup norm on the circle of radius \( R \) . Suppose that there exists a constant \( C \) and a positive integer \( k \) such that\n\n\[ \parallel f{\parallel }_{R} \leqq C{R}^{k} \]\n\nfor arbitrarily large \( R \) . Then ... | Proof. Exercise 3, but we carry out one important special case explicitly: | No |
Theorem 7.5 (Liouville's Theorem). A bounded entire function is constant. | Proof. If \( f \) is bounded, then \( \parallel f{\parallel }_{R} \) is bounded for all \( R \) . In the preceding theorem, we let \( R \) tend to infinity, and conclude that the coefficients are all equal to 0 if \( n \geqq 1 \) . This proves Liouville’s theorem. | No |
Corollary 7.6. A polynomial over the complex numbers which does not have a root in \( \mathbf{C} \) is constant. | Proof. Let \( f\left( z\right) \) be a non-constant polynomial,\n\n\[ f\left( z\right) = {a}_{n}{z}^{n} + \cdots + {a}_{0}, \]\n\nwith \( {a}_{n} \neq 0 \) . Suppose that \( f\left( z\right) \neq 0 \) for all \( z \) . Then the function\n\n\[ g\left( z\right) = 1/f\left( z\right) \]\n\nis defined for all \( z \) and an... | Yes |
Theorem 7.7. Let \( \gamma \) be a path in an open set \( U \) and let \( g \) be a continuous function on \( \gamma \) (i.e. on the image \( \gamma \left( \left\lbrack {a, b}\right\rbrack \right) \) if \( \gamma \) is defined on \( \left\lbrack {a, b}\right\rbrack ) \) . If \( z \) is not on \( \gamma \), define\n\n\[... | Proof. Let \( {z}_{0} \in U \) and \( {z}_{0} \) not on \( \gamma \) . Since the image of \( \gamma \) is compact, there is a minimum distance between \( {z}_{0} \) and points on \( \gamma \) . Select \( 0 < R < \operatorname{dist}\left( {{z}_{0},\gamma }\right) \), and take \( R \) also small enough that the closed di... | Yes |
Corollary 7.8. Let \( f \) be analytic on a closed disc \( \bar{D}\left( {{z}_{0}, R}\right), R > 0 \) . Let\n\n\( 0 < {R}_{1} < R \) . Denote by \( \parallel f{\parallel }_{R} \) the sup norm of \( f \) on the circle of radius\n\nR. Then for \( z \in \bar{D}\left( {{z}_{0},{R}_{1}}\right) \) we have\n\n\[\n\left| {{f}... | Proof. This is immediate by using Theorem 7.1, and putting \( g = f \) inside the integral, with a factor of \( 1/{2\pi i} \) in front. The factor \( R \) in the numerator comes from the length of the circle in the integral. The \( {2\pi } \) in the denominator cancels the \( {2\pi } \) in the numerator, coming from th... | Yes |
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