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Corollary 20.18 Let \( D \) be a connected digraph and \( \mathbf{B} \) and \( \mathbf{C} \) unimodular basis matrices of \( \mathcal{B} \) and \( \mathcal{C} \), respectively. Then\n\n\[ t\left( G\right) = \pm \det \left\lbrack \begin{array}{l} \mathbf{B} \\ \mathbf{C} \end{array}\right\rbrack \] | Proof By Theorems 20.14 and 20.17,\n\n\[ {\left( t\left( G\right) \right) }^{2} = \det {\mathbf{{BB}}}^{t}\det {\mathbf{{CC}}}^{t} = \det \left\lbrack \begin{matrix} {\mathbf{{BB}}}^{t} & \mathbf{0} \\ \mathbf{0} & {\mathbf{{CC}}}^{t} \end{matrix}\right\rbrack \]\n\nBecause \( \mathcal{B} \) and \( \mathcal{C} \) are o... | Yes |
Theorem 20.19 Let \( D \mathrel{\text{:=}} D\left( {x, y}\right) \) be a connected directed graph. For any real number \( \mathrm{i} \), there exists a unique current flow in \( D \) of value \( \mathrm{i} \) from \( x \) to \( y \) . | Proof Let \( \mathbf{K} = {\mathbf{M}}_{y} \) be a Kirchhoff matrix of \( D \) . We assume that the first row of \( \mathbf{K} \) is indexed by \( x \) . By definition, a function \( f : A \rightarrow \mathbb{R} \) is a current flow of value i from \( x \) to \( y \) if it satisfies the two systems of equations:\n\n\[ ... | Yes |
Consider the planar graph \( G \) in Figure 20.6a. On deleting the edge \( {xy} \) and orienting each remaining edge, as shown, we obtain the digraph \( D \) of Figure 20.6b. It can be checked that \( t\left( D\right) = {66} \) (Exercise 20.4.4). By considering the tree \( T \mathrel{\text{:=}} \left\{ {{a}_{1},{a}_{2}... | The solution to this system of equations is given by\n\n\[ \left( {{f}_{1},{f}_{2},{f}_{3},{f}_{4},{f}_{5},{f}_{6},{f}_{7},{f}_{8},{f}_{9}}\right) = \left( {{36},{30},{14},{16},{20},2,{18},{28},8}\right) \] | Yes |
Theorem 20.21 In any directed graph \( D \mathrel{\text{:=}} D\left( {x, y}\right) \), the functions \( f \) and \( g \) defined by (20.14) and (20.15) are equal. This function is therefore the unique current flow in \( D \) of value \( t\left( D\right) \) . | Proof Let \( T \) be a spanning tree of \( D \), and \( a \) an arc of the path \( P \mathrel{\text{:=}} {xTy} \) . Consider the spanning tree \( {T}^{\prime } \mathrel{\text{:=}} \left( {T \smallsetminus a}\right) + {a}^{\prime } \) of \( {D}^{\prime } \) . Then the arc \( a \) is a forward arc of \( P \) if it belong... | Yes |
Theorem 20.22 The effective resistance between \( x \) and \( y \) in an electrical network \( D\left( {x, y}\right) \) is given by the formula\n\n\[ \n{\mathrm{r}}_{xy} = \frac{\det {\mathbf{{LL}}}^{\mathbf{t}}}{\det {\mathbf{{KK}}}^{\mathbf{t}}} \n\] | Proof Add a new arc \( {a}^{\prime } \mathrel{\text{:=}} \left( {y, x}\right) \) to \( D \), so as to obtain a digraph \( {D}^{\prime } \) . By Theorem 20.21, \( g \mathrel{\text{:=}} \mathop{\sum }\limits_{{T}^{\prime }}{g}_{{T}^{\prime }} \) is the unique current flow of value \( t\left( D\right) \) . For any \( {xy}... | Yes |
Corollary 20.23 If \( x \) and \( y \) are adjacent vertices of a digraph \( D \) , | \[ {\mathrm{r}}_{xy} = \frac{{t}_{xy}\left( D\right) }{t\left( D\right) } \] Proof As in the proof of Theorem 20.22, one has \( {t}_{xy}\left( D\right) = \det {\mathbf{{LL}}}^{\mathbf{t}} \) . Also, \( t\left( D\right) = \) \( \det {\mathbf{{KK}}}^{\mathbf{t}} \) . | No |
What is the probability that a drunkard eventually returns home, assuming that he takes a random walk on the two-dimensional integer lattice \( {\mathbb{Z}}^{2} \)? | Pólya (1921) proved that this probability is one, in other words, that the drunkard is sure to get back home eventually (despite his inebriated state). | Yes |
Theorem 20.25 Let \( x \) and \( y \) be distinct vertices of a simple connected graph \( G \) . The probability \( {P}_{x} \) that a random \( x \) -walk on \( G \) hits \( y \) before returning to \( x \) is given by\n\n\[ \n{P}_{x} = \frac{1}{d\left( x\right) {\mathrm{r}}_{xy}} \n\] | Proof For \( v \in V \smallsetminus \{ x\} \), denote by \( {P}_{v} \) the probability that a random \( v \) -walk on \( G \) hits \( y \) before hitting \( x \) . Then \( {P}_{y} = 1 \), and\n\n\[ \n{P}_{v} = \frac{1}{d\left( v\right) }\mathop{\sum }\limits_{w}{P}_{w},\;v \in V \smallsetminus \{ y\} \n\]\n\nthat is,\n... | Yes |
Corollary 20.27 The cover time \( C \) of a graph \( G \) is at most \( {2m}\left( {n - 1}\right) \) . | Proof Let \( T \) be any spanning tree of \( G \), and let \( \left( {v = {v}_{0},{v}_{1},\ldots ,{v}_{{2n} - 2} = v}\right) \) be the sequence of vertices encountered in a walk around \( T \) (not a random walk) which starts at an arbitrary vertex \( v \) and traverses each edge of \( T \) once in each direction. Cons... | Yes |
Proposition 21.1 A digraph \( D \) is \( k \) -vertex-colourable if and only if it admits a nowhere-zero tension over \( {\mathbb{Z}}_{k} \) . | Proof Firstly, suppose that \( D \) has a proper \( k \) -vertex-colouring \( c : V \rightarrow {\mathbb{Z}}_{k} \) . Consider the tension \( g : A \rightarrow {\mathbb{Z}}_{k} \) defined by \( g\left( a\right) \mathrel{\text{:=}} c\left( u\right) - c\left( v\right) \) for each arc \( a \mathrel{\text{:=}} \) \( \left(... | No |
Theorem 21.2 A plane digraph \( D \) is \( k \) -face-colourable if and only if it admits a nowhere-zero circulation over \( {\mathbb{Z}}_{k} \) . | Proof By the analogue of Theorem 20.5 for circulations and tensions over \( {\mathbb{Z}}_{k} \), a function \( f : A\left( D\right) \rightarrow {\mathbb{Z}}_{k} \) is a circulation in \( D \) over \( {\mathbb{Z}}_{k} \) if and only if the corresponding function \( {f}^{ * } : A\left( {D}^{ * }\right) \rightarrow {\math... | Yes |
Theorem 21.5 A 2-edge-connected cubic graph admits a 3-flow if and only if it is bipartite. | Proof Let \( G \mathrel{\text{:=}} G\left\lbrack {X, Y}\right\rbrack \) be a bipartite cubic graph. By Theorem 17.2, \( G \) is 3- edge-colourable, so there exist three disjoint perfect matchings \( {M}_{1},{M}_{2} \), and \( {M}_{3} \) in \( G \) such that \( E = {M}_{1} \cup {M}_{2} \cup {M}_{3} \) . Orient the edges... | Yes |
Theorem 21.6 For any graph \( G \), any link \( e \) of \( G \), and any finite additive abelian group \( \Gamma \) ,\n\n\[ F\left( {G,\Gamma }\right) = F\left( {G/e,\Gamma }\right) - F\left( {G \smallsetminus e,\Gamma }\right) \] | By a simple inductive argument similar to the one used to demonstrate that the number of \( k \) -colourings is a polynomial in \( k \), one can derive the following implication of Theorem 21.6 (Exercise 21.2.4). What is striking here is that \( F\left( {G,\Gamma }\right) \) depends not on the structure of the group \(... | No |
Theorem 21.7 For any graph \( G \) without cut edges, there exists a polynomial \( Q\left( {G, x}\right) \) such that \( F\left( {G,\Gamma }\right) = Q\left( {G, k}\right) \) for every additive abelian group \( \Gamma \) of order \( k \) . Moreover, if \( G \) is simple and \( e \) is any edge of \( G \), then \( Q\lef... | \[ Q\left( {G, x}\right) = Q\left( {G/e, x}\right) - Q\left( {G \smallsetminus e, x}\right) \] | Yes |
Theorem 21.9 Let \( G \) be a graph and let \( {k}_{1} \) and \( {k}_{2} \) be integers, where \( {k}_{i} \geq 2 \) , \( i = 1,2 \) . Then \( G \) admits a \( {k}_{1}{k}_{2} \) -flow if and only if \( G = {G}_{1} \cup {G}_{2} \), where \( {G}_{i} \) admits \( a{k}_{i} \) -flow, \( i = 1,2 \) . | Proof If \( G \) has a \( {k}_{1}{k}_{2} \) -flow, then it has a nowhere-zero circulation over \( {\mathbb{Z}}_{{k}_{1}{k}_{2}} \) , by virtue of Theorem 21.3. By Corollary 21.8, this implies that \( G \) has a nowhere-zero circulation \( f \mathrel{\text{:=}} \left( {{f}_{1},{f}_{2}}\right) \) over \( {\mathbb{Z}}_{{k... | Yes |
Corollary 21.10 A graph admits a \( {2}^{k} \) -flow if and only if it admits a covering by \( k \) even subgraphs. | Proof Apply Theorem 21.9 recursively, with \( {k}_{i} = 2,1 \leq i \leq k \), and invoke Theorem 21.4. | No |
Theorem 21.12 Every graph which admits an orientable double cover by \( k \) even subgraphs admits a \( k \) -flow. | Proof Let \( \left\{ {{C}_{i} : 1 \leq i \leq k}\right\} \) be an orientable double cover of \( G \) by \( k \) even subgraphs and let \( {f}_{i} \) be the positive 2-flow on an even orientation \( {D}_{i} \) of \( {C}_{i} \) . Now consider a fixed orientation \( D \) of \( G \) and, for \( 1 \leq i \leq k \), let \( {... | Yes |
Every \( {2k} \) -edge-connected graph contains \( k \) edge-disjoint spanning trees. | Let \( G \) be a \( {2k} \) -edge-connected graph and let \( \mathcal{P} \mathrel{\text{:=}} \left\{ {{V}_{1},{V}_{2},\ldots ,{V}_{p}}\right\} \) be a partition of \( V \) . The number edges from \( {V}_{i} \) to the other parts of \( \mathcal{P} \) is \( d\left( {V}_{i}\right) \) and, since \( G \) is \( {2k} \) -edge... | Yes |
Theorem 21.19 Every 4-edge-connected graph admits a covering by two even subgraphs. | Proof Let \( G \) be a 4-edge-connected graph. By Corollary 21.18, \( G \) has two edge-disjoint spanning trees, hence (Exercise 4.3.10) a covering by two even subgraphs. | No |
Theorem 21.21 Every 2-edge-connected graph admits a covering by three even subgraphs. | Proof It suffices to prove the assertion for 3-edge-connected graphs (Exercise 21.5.1). Thus let \( G \) be a 3-edge-connected graph. Denote by \( H \) the graph obtained by duplicating each edge of \( G \) . Being 6-edge-connected, \( H \) has three edge-disjoint spanning trees, by Corollary 21.18. These trees corresp... | No |
Proposition 21.23 If a graph admits a covering by \( k \) even subgraphs, then it admits a \( {2}^{k - 1} \) -cover by \( {2}^{k} - 1 \) even subgraphs. | Proof Let \( \left\{ {{C}_{1},{C}_{2}\ldots ,{C}_{k}}\right\} \) be a covering of a graph \( G \) by \( k \) even subgraphs, and let \( e \) be an edge of \( G \) that belongs to \( j \) of these subgraphs, without loss of generality \( {C}_{1},{C}_{2},\ldots ,{C}_{j} \). Then \( e \) belongs to all the even subgraphs ... | Yes |
Lemma 21.26 Let \( S \) be a set of edges of a graph \( G \) whose 2-closure is the entire set \( E \) . Then there exists a circulation in \( G \) over \( {\mathbb{Z}}_{3} \) whose support includes \( E \smallsetminus S \) . | Proof The proof is by induction on \( \left| {E \smallsetminus S}\right| \), the result being trivial when \( S = E \) . Assume that \( S \) is a proper subset of \( E \) . By hypothesis, there is a cycle \( C \) in \( G \) such that \( 1 \leq \left| {C \smallsetminus S}\right| \leq 2 \) . Set \( {S}^{\prime } \mathrel... | No |
Lemma 21.27 Every 3-edge-connected graph \( G \) contains an even subgraph \( C \) whose 2-closure is \( E \) . | Proof Let \( C \) be an even subgraph of \( G \) such that:\ni) the subgraph \( H \) of \( G \) induced by the 2-closure of \( C \) is connected,\nii) subject to (i), \( C \) is as large as possible.\n\nWe may assume that \( H \) is not a spanning subgraph of \( G \) . Otherwise, by the definition of 2-closure, \( H = ... | Yes |
Every 2-edge-connected graph admits a 6-flow. | Proof By Exercise 21.3.5b, it suffices to prove the theorem for 3-edge-connected graphs. Let \( G \) be such a graph. By Lemma 21.27, \( G \) contains an even subgraph \( C \) whose 2-closure is \( E \) . Let \( D \) be an orientation of \( G,{f}_{1} \) a circulation in \( D \) over \( {\mathbb{Z}}_{2} \) whose support... | No |
Theorem 21.29 Every 2-edge-connected graph admits a sextuple cover by ten even subgraphs. | Proof It suffices to prove the theorem for 3-edge-connected graphs (Exercise 21.6.4). Let \( G \) be such a graph. By Lemma 21.27, \( G \) contains an even subgraph \( C \) whose edge set has 2-closure \( E \) . Consider an orientation \( D \) of \( G \) whose restriction to \( C \) is even, and let \( {f}^{\prime } \)... | No |
Theorem 21.31 The Tutte polynomial \( T\left( {G;x, y}\right) \) has the following properties. | \( \vartriangleright \;T\left( {{L}_{0};x, y}\right) = 1,\;T\left( {{B}_{1};x, y}\right) = x,\;T\left( {{L}_{1};x, y}\right) = y. \)\n\n\( \vartriangleright \) If \( e \) is a loop of \( G \) ,\n\n\[ T\left( {G;x, y}\right) = y \cdot T\left( {G \smallsetminus e;x, y}\right) \]\n\n\( \vartriangleright \) If \( e \) is a... | Yes |
Theorem 1.1. Let \( D \subseteq \mathbb{C} \) denote a domain (an open connected set), and let \( f = u + {\iota v} : D \rightarrow \mathbb{C} \) be a complex-valued function defined on \( D \) . The following conditions are equivalent (here \( u \) and \( v \) are real-valued functions of the complex variable \( z = x... | ## 1.3 The Plan for the Proof\n\nWe prove the fundamental theorem by showing the following implications.\n\n\[ \n\left( 1\right) \Leftrightarrow \left( 2\right) \Rightarrow \left( 3\right) \Rightarrow \left( 4\right) \Rightarrow \left( 5\right) \Rightarrow \left( 6\right) \Rightarrow \left( 1\right) ; \n\]\n\n\[ \n\lef... | No |
Theorem 2.2. If \( \\left\\{ {z}_{n}\\right\\} \) and \( \\left\\{ {w}_{n}\\right\\} \) are Cauchy sequences of complex numbers, then\n\n(a) \( \\left\\{ {{z}_{n} + \\alpha {w}_{n}}\\right\\} \) is Cauchy for all \( \\alpha \\in \\mathbb{C} \) . | Proof. (a) It suffices to assume that \( \\alpha \\neq 0 \) . Given \( \\epsilon > 0 \), choose \( {N}_{1} \) such that \( \\left\\| {{z}_{n} - {z}_{m}}\\right\\| < \\frac{\\epsilon }{2} \) for all \( n, m > {N}_{1} \) and choose \( {N}_{2} \) such that \( \\left\\| {{w}_{n} - {w}_{m}}\\right\\| < \\frac{\\epsilon }{2\... | Yes |
Corollary 2.5. \( \left( {\mathbb{C}, d}\right) \) is a complete metric space; that is, every Cauchy sequence of complex numbers converges to a complex number: | Proof. Observe that the metric on \( \mathbb{C} \) restricts to the Euclidean metric on \( \mathbb{R} \), which is complete, and applies the previous corollary. | No |
Theorem 2.20. Let \( K \subset \mathbb{C} \) be a compact set and \( f : K \rightarrow \mathbb{C} \) be a continuous function on \( K \) . Then \( f \) is uniformly continuous on \( K \) . | Proof. A continuous mapping from a compact metric space to a metric space is uniformly continuous. | Yes |
Theorem 2.23. Let \( \left\{ {f}_{n}\right\} \) be a sequence of functions defined on \( S \subseteq \mathbb{C} \) . If:\n\n(1) \( \left\{ {f}_{n}\right\} \) converges uniformly on \( S \) .\n\n(2) Each \( {f}_{n} \) is continuous on \( S \) .\n\nThen the function \( f \) defined by\n\n\[ f\left( z\right) = \mathop{\li... | Proof. Start with two points \( z \) and \( c \) in \( S \) . Then for each natural number \( n \) we have\n\n\[ \left| {f\left( z\right) - f\left( c\right) }\right| \leq \left| {f\left( z\right) - {f}_{n}\left( z\right) }\right| + \left| {{f}_{n}\left( z\right) - {f}_{n}\left( c\right) }\right| + \left| {{f}_{n}\left(... | Yes |
Theorem 2.33. If \( f = u + {vv} \) is differentiable at \( c = a + {vb} \), then \( u \) and \( v \) have partial derivatives with respect to \( x \) and \( y \) at \( c \), and they satisfy the Cauchy-Riemann equations:\n\n\[ \n{u}_{x}\left( {a, b}\right) = {v}_{y}\left( {a, b}\right) ,\;{u}_{y}\left( {a, b}\right) =... | Proof. First take \( h = \alpha \), with \( \alpha \) real, in the limit (2.9) appearing in the definition of differentiability and compute\n\n\[ \n{f}^{\prime }\left( c\right) = {u}_{x}\left( {a, b}\right) + \iota {v}_{x}\left( {a, b}\right) .\n\]\n\nThen take \( h = \imath \beta \), with \( \beta \) real, and compute... | Yes |
Theorem 2.40. If the function \( f \) has continuous first partial derivatives in a neighborhood of \( c \) that satisfy the \( \mathrm{{CR}} \) equations at \( c \), then \( f \) is (complex) differentiable at \( c \) . | Proof. The theorem is an immediate consequence of (2.12), since in this case \( {f}_{\bar{z}}\left( c\right) = 0 \) and hence \( {f}^{\prime }\left( c\right) = {f}_{z}\left( c\right) \) . | Yes |
Theorem 2.43. If \( f \) is holomorphic and real-valued on a domain \( D \), then \( f \) is constant. | Proof. As usual we write \( f = u + {uv} \) ; in this case \( v = 0 \) . The CR equations say \( {u}_{x} = {v}_{y} = 0 \) and \( {u}_{y} = - {v}_{x} = 0 \) . Thus \( u \) is constant, since \( D \) is connected. | Yes |
Theorem 2.44. If \( f \) is holomorphic and \( {f}^{\prime } = 0 \) on a domain \( D \), then \( f \) is constant. | Proof. As above \( f = u + {\iota v} \) and \( {f}^{\prime } = {u}_{x} + \iota {v}_{x} = 0 \) . The last equation together with the CR equations say \( 0 = {u}_{x} = {v}_{y} \) and; \( 0 = {v}_{x} = - {u}_{y} \) . Thus both \( u \) and \( v \) are constant, since \( D \) is connected. | Yes |
Theorem 3.5 (Weierstrass \( M \) -test). Normal convergence implies uniform and absolute convergence. | Proof. With notation as in the definition of normal convergence, if \( {N}_{1} < N \) are positive integers, then\n\n\[ \left| {{S}_{N}\left( z\right) - {S}_{{N}_{1}}\left( z\right) }\right| \leq \mathop{\sum }\limits_{{n = {N}_{1} + 1}}^{N}{M}_{n}\text{ for all }z \in B \]\n\n(needed for the uniform convergence argume... | Yes |
Lemma 3.7 (Abel’s Lemma). Assume that \( 0 < r < {r}_{0} \) . If there exists a positive number \( M \) such that\n\n\[ \left| {{a}_{n}{r}_{0}^{n}}\right| \leq M\text{ for all }n \in {\mathbb{Z}}_{ > 0}, \]\n\nthen the series \( \sum {a}_{n}{z}^{n} \) converges normally for all \( z \) with \( \left| z\right| \leq r \)... | Proof. For all \( \left| z\right| \leq r \) we have\n\n\[ \left| {{a}_{n}{z}^{n}}\right| = \left| {a}_{n}\right| {\left| z\right| }^{n} \leq \left| {a}_{n}\right| {\left| r\right| }^{n} = \left| {a}_{n}\right| {\left( \frac{r}{{r}_{0}}\right) }^{n}{r}_{0}^{n} \leq M{\left( \frac{r}{{r}_{0}}\right) }^{n}. \]\n\nSince \(... | Yes |
Theorem 3.9. Let \( \sum {a}_{n}{z}^{n} \) be a power series with radius of convergence \( \rho > 0 \) . Then\n\n(a) For any \( 0 < r < \rho \), the series \( \sum {a}_{n}{z}^{n} \) converges normally, absolutely, and uniformly for \( \left| z\right| \leq r \) .\n\n(b) The series \( \sum {a}_{n}{z}^{n} \) diverges for ... | Proof. (a) For any \( {r}_{0} \) satisfying \( r < {r}_{0} < \rho \), the series \( \sum \left| {a}_{n}\right| {r}_{0}^{n} \) converges. Then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\left| {a}_{n}\right| {r}_{0}^{n} = 0 \), and thus there exists an \( M > 0 \) with \( \left| {a}_{n}\right| {r}_{0}^{n} \leq M ... | Yes |
Corollary 3.10. Let \( \sum {a}_{n}{z}^{n} \) be a power series with radius of convergence \( \rho > 0 \) . Then the function defined by \( S\left( z\right) = \sum {a}_{n}{z}^{n} \) is continuous for \( \left| z\right| < \rho \) . | Proof. It follows immediately from Theorems 2.23 and 3.9. | No |
Let \( {u}_{n} = \sin \left( \frac{n\pi }{2}\right), n = 0,1,2,\ldots \) ; this is the sequence: \( \{ 0,1,0 \) , \( - 1,\ldots \} \) . Hence for all \( p \in {\mathbb{Z}}_{ \geq 0} \) we have | \[ {a}_{p} = \mathop{\sup }\limits_{{n \geq p}}{u}_{n} = 1\text{and thus}\mathop{\limsup }\limits_{n}{u}_{n} = \mathop{\lim }\limits_{{p \rightarrow \infty }}{a}_{p} = 1\text{, and} \] \[ {b}_{p} = \mathop{\inf }\limits_{{n \geq p}}{u}_{n} = - 1\text{and thus}\mathop{\liminf }\limits_{n}{u}_{n} = \mathop{\lim }\limits_... | Yes |
Consider the series \( \mathop{\sum }\limits_{{n \geq 1}}{n}^{-s} \), with \( s \) a positive real number. | Using standard calculus techniques, one shows that\n\n\[\n\mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{{\left( n + 1\right) }^{-s}}{{n}^{-s}} = 1 = \mathop{\lim }\limits_{{n \rightarrow \infty }}{\left( {n}^{-s}\right) }^{\frac{1}{n}};\n\]\n\nthus both the ratio and the root tests are inconclusive in this case. | No |
Theorem 3.16 (Cauchy-Hadamard). The radius of convergence \( \rho \) of the power series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{z}^{n} \) is given by \[ \frac{1}{\rho } = \mathop{\lim }\limits_{n}{\left| {a}_{n}\right| }^{\frac{1}{n}} \] | Proof. Let \( L = \mathop{\lim }\limits_{n}{\left| {a}_{n}\right| }^{\frac{1}{n}} \) . Thus \( \mathop{\lim }\limits_{n}{\left| {a}_{n}{r}^{n}\right| }^{\frac{1}{n}} = {rL} \) for all \( r \geq 0 \), and we conclude by the root test that the associated series \( \sum \left| {a}_{n}\right| {r}^{n} \) converges for \( 0 ... | Yes |
Corollary 3.21. If \( S\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{z}^{n} \) for \( \left| z\right| < \rho \), then for all \( n \in {\mathbb{Z}}_{ \geq 0} \) and all \( \left| z\right| < \rho \), the power series for the derivatives \( {S}^{\left( n\right) } = \frac{{d}^{n}S}{\mathrm{\;d}{z}^{n}... | Proof. Applying the theorem and induction on \( n \) shows that\n\n\[ \n{S}^{\left( n\right) }\left( z\right) = n!{a}_{n} + \frac{\left( {n + 1}\right) !}{1!}{a}_{n + 1}z + \cdots = \mathop{\sum }\limits_{{k = n}}^{\infty }\frac{k!}{\left( {n - k}\right) !}{a}_{k}{z}^{n - k}\n\]\n\nfor all \( \left| z\right| < \rho \) ... | Yes |
Theorem 3.22 (Abel’s Limit Theorem). Assume that the power series \( \sum {a}_{n}{z}^{n} \) has finite radius of convergence \( \rho > 0 \) . If \( \sum {a}_{n}{z}_{0}^{n} \) converges for some \( {z}_{0} \) with \( \left| {z}_{0}\right| = \) \( \rho \), then \( f\left( z\right) = \sum {a}_{n}{z}^{n} \) is defined for ... | Proof. By the change of variable \( w = \frac{z}{{z}_{0}} \) we may assume that \( \rho = 1 = {z}_{0} \) (replace \( \left. {{a}_{n}\text{by}{a}_{n}{z}_{0}^{n}}\right) \) . Thus \( \sum {a}_{n} \) converges to \( f\left( 1\right) \) . By changing \( {a}_{0} \) to \( {a}_{0} - f\left( 1\right) \), we may assume that \( ... | Yes |
Proposition 3.26. Let \( c \in \mathbb{C} \) . The function \( f\left( z\right) = c{\mathrm{e}}^{z} \) is the unique power series, and also the unique entire function, satisfying\n\n\[ \n{f}^{\prime }\left( z\right) = f\left( z\right) \text{ and }f\left( 0\right) = c.\n\] | Proof. It is trivial that \( z \mapsto c{\mathrm{e}}^{z} \) satisfies (3.5) and is the unique power series to do so; we already know that this is an entire function. We postpone the proof that this is the unique entire function that satisfies (3.5) until after we establish the next two propositions. | No |
Proposition 3.27. For all \( z \in \mathbb{C} \), \[ {\mathrm{e}}^{z}{\mathrm{e}}^{-z} = 1 \] Thus \[ {\mathrm{e}}^{z} \neq 0\text{ for all }z \in \mathbb{C}. \] | Proof. Set \( h\left( z\right) = {\mathrm{e}}^{z}{\mathrm{e}}^{-z} \) for all \( z \in \mathbb{C} \). Then \( h \) is an entire function, and the rules for differentiation tell us that \[ {h}^{\prime }\left( z\right) = {\mathrm{e}}^{z}{\mathrm{e}}^{-z} - {\mathrm{e}}^{z}{\mathrm{e}}^{-z} = 0 \] for all \( z \). Therefo... | Yes |
Proposition 3.28. \( {\mathrm{e}}^{z + c} = {\mathrm{e}}^{z}{\mathrm{e}}^{c} \) for all \( z \) and \( c \) in \( \mathbb{C} \) . | Proof. Define \( h\left( z\right) = \frac{{\mathrm{e}}^{z + c}}{{\mathrm{e}}^{c}} \), with \( c \) fixed in \( \mathbb{C} \) . The function \( h \) has a power series expansion that converges for all \( z \in \mathbb{C},{h}^{\prime }\left( z\right) = h\left( z\right) \) for all \( z \in \mathbb{C} \), and \( h\left( 0\... | Yes |
Proposition 3.29. \( \overline{{\mathrm{e}}^{z}} = {\mathrm{e}}^{\bar{z}} \) for all \( z \in \mathbb{C} \) . | Proof. This follows directly from the definition (3.4) of the exponential function. | No |
Proposition 3.30. For any \( z \) in \( \mathbb{C} \), write \( z = x + {vy} \), with \( x \) and \( y \) in \( \mathbb{R} \). Then\n\n\[ \n{\left| {\mathrm{e}}^{iy}\right| }^{2} = {\mathrm{e}}^{iy}{\mathrm{e}}^{-{iy}} = 1 \n\]\n\nand thus\n\n\[ \n\left| {\mathrm{e}}^{z}\right| = {\mathrm{e}}^{x} \n\] | The exponential function leads us immediately to our next section, the complex trigonometric functions. | No |
Theorem 3.36. Let \( D \) be a domain in \( \mathbb{C} \) with \( 0 \notin D \) . If \( f \) is a branch of the logarithm on \( D \), then \( g \) is also a branch of the logarithm in \( D \) if and only if there is an \( n \in \mathbb{Z} \) such that \( g\left( z\right) = f\left( z\right) + {2\pi }\imath n \) for all ... | Proof. If \( g = f + {2\pi }\imath n \) with \( n \in \mathbb{Z} \), then \( {\mathrm{e}}^{g\left( z\right) } = {\mathrm{e}}^{f\left( z\right) }{\mathrm{e}}^{{2\pi }\imath n} = z \) for all \( z \) in \( D \) . For a proof of the converse, define \[ h\left( z\right) = \frac{f\left( z\right) - g\left( z\right) }{{2\pi }... | Yes |
Corollary 3.37. Every branch of the logarithm on a domain \( D \) (with \( 0 \notin D \) ) is holomorphic on \( D \) . | Proof. Holomorphicity is a local property, and there exist holomorphic branches of the logarithm in every sufficiently small disc that does not contain the origin. | No |
Theorem 3.38. For \( z \in \mathbb{C} \) with \( \left| z\right| < 1 \) , \[ \log \left( {1 + z}\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( -1\right) }^{n - 1}\frac{{z}^{n}}{n} = z - \frac{{z}^{2}}{2} + \frac{{z}^{3}}{3} - \cdots . \] | Proof. We first compute the radius of convergence of the given series using the ratio test: \( \frac{1}{\rho } = \mathop{\lim }\limits_{n}\left| \frac{n}{n + 1}\right| = 1 \) . Thus the function defined by \[ f\left( z\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( -1\right) }^{n - 1}\frac{{z}^{n}}{n} \] (3.... | Yes |
Theorem 3.41. Let \( f \) be a function defined in a neighborhood of \( c \in \mathbb{C} \) that has a power series expansion at \( c \) with radius of convergence \( \rho > 0 \) . Then\n\n(a) \( f \) is holomorphic and \( {\mathbf{C}}^{\infty } \) in a neighborhood of \( c \) .\n\n(b) If \( g \) also has a power serie... | Proof. Without loss of generality we assume \( c = 0 \) :\n\n(a) Already verified in Theorem 3.19.\n\n(b) For some \( r > 0 \), we have\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{z}^{n}\text{ and }g\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{b}_{n}{z}^{n}\text{ for all }\l... | Yes |
Lemma 3.42. If \( S\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{z}^{n} \) has radius of convergence \( \rho > 0 \), then for any \( c \in \mathbb{C} \) with \( \left| c\right| < \rho \), the power series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{S}^{\left( n\right) }\left( c\right) }{n!}{... | Proof. Set \( R = \left| c\right| < \rho \) (see Fig. 3.2). The argument consists of two steps:\n\n(I) We show first that \( \mathop{\sum }\limits_{{p = 0}}^{\infty }\frac{{S}^{\left( p\right) }\left( c\right) }{p!}{w}^{p} \) is absolutely convergent for \( \left| w\right| < \rho - R \) .\n\nWe know from Corollary 3.21... | Yes |
We study the holomorphic function\n\n\[ S\\left( z\\right) = \\frac{1}{1 - z}, z \\in {\\mathbb{C}}_{ \\neq 1} \]\n\n it satisfies\n\n\[ S\\left( z\\right) = 1 + z + {z}^{2} + \\cdots \\text{ for }\\left| z\\right| < 1 \]\nthat is, \( S \) has a power series expansion at the origin with radius of convergence \( \\rho =... | Let us take \( c = - \\frac{1}{2} \) . Then \( {S}^{\\left( p\\right) }\\left( z\\right) = p!{\\left( 1 - z\\right) }^{-1 - p} \) and thus \( {S}^{\\left( p\\right) }\\left( {-\\frac{1}{2}}\\right) = \) \( p!{\\left( \\frac{2}{3}\\right) }^{1 + p} \) . A calculation shows that the power series \( \\mathop{\\sum }\\limi... | Yes |
Theorem 3.45. Let \( f \) be a function defined on a domain \( D \) in \( \mathbb{C} \). Assume that \( f \) has a power series expansion at each point of \( D \), and let \( c \in D \). The following conditions are equivalent:\n\n(a) \( {f}^{\left( n\right) }\left( c\right) = 0 \) for \( n = 0,1,2,\ldots \).\n\n(b) \(... | Proof. First note that there are obvious implications: \( \left( a\right) \Leftrightarrow \left( b\right) \) and \( \left( d\right) \Rightarrow \left( b\right) \Rightarrow \) \( \left( c\right) \) . To complete the proof, we will now show that \( \left( c\right) \Rightarrow \left( a\right) \) and that \( \left( a\right... | Yes |
Theorem 3.55. If a function \( f \) has a power series expansion at \( c \) and \( f\left( c\right) \neq 0 \) , then \( \frac{1}{f} \) also has a power series expansion at \( c \) . | Proof. Without loss of generality we assume \( c = 0 \) and \( f\left( 0\right) = 1 \) . Thus\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{z}^{n},{a}_{0} = 1, \]\n\nand the radius of convergence of the series is nonzero. We want to find the reciprocal power series, that is, a series \( g \)... | Yes |
Corollary 3.63. If \( D \) be a domain and \( c \in D \), then\n\n\[ \n{v}_{c} : \mathbf{M}{\left( D\right) }_{ \neq 0} \rightarrow \mathbb{Z} \]\n\nis a homomorphism; that is, \( {v}_{c}\left( {f \cdot g}\right) = {v}_{c}\left( f\right) + {v}_{c}\left( g\right) \) for all \( f \) and \( g \) in \( \mathbf{M}{\left( D\... | Defining \( {v}_{c}\left( 0\right) = + \infty \), we also have\n\n\[ \n{v}_{c}\left( {f + g}\right) \geq \min \left\{ {{v}_{c}\left( f\right) ,{v}_{c}\left( g\right) }\right\} \text{ for all }f\text{ and }g\text{ in }\mathbf{M}\left( D\right) ; \]\n\nthat is, \( {v}_{c} \) is a (discrete) valuation \( {}^{10} \) (of ra... | No |
Corollary 3.67. If \( f \in \mathbf{M}\left( D\right) \), then \( {f}^{\prime } \in \mathbf{M}\left( D\right) \). If in addition \( {v}_{c}\left( f\right) \neq 0 \) for \( c \in D \), then | \[ {v}_{c}\left( {f}^{\prime }\right) = {v}_{c}\left( f\right) - 1 \] | No |
Lemma 4.11. If \( D \) is a domain in \( \mathbb{C} \), then any two points in \( D \) can be joined by a pdp in \( D \) . | Proof. Fix \( c \in D \) and let\n\n\[ E = \{ z \in D;z\text{ can be joined to }c\text{ by a pdp in }D\} . \]\n\nThe set \( E \) is open in \( D \), because if \( z \) denotes any point in \( E \), then, since \( D \) is an open set, there is a small disc \( U \) with center at \( z \) contained in \( D \), and any poi... | Yes |
Lemma 4.14. Let \( \omega \) be a differential form on a domain \( D \) . Then \( \omega \) is exact on \( D \) if and only if \( {\int }_{\gamma }\omega = 0 \) for all closed pdps \( \gamma \) in \( D \) . | Proof. Assume that \( \omega \) is exact. Then there exists a \( {\mathbf{C}}^{1} \) -function \( F \) on \( D \) with\n\n\[ \omega = {F}_{x}\mathrm{\;d}x + {F}_{y}\mathrm{\;d}y. \]\n\nIf \( \gamma \) is a pdp parameterized by \( \left\lbrack {a, b}\right\rbrack \) joining two points \( {P}_{1} \) to \( {P}_{2} \) in \... | Yes |
Corollary 4.15 (of proof). If \( \mathrm{d}F \) is an exact differential form on the domain \( D \) and \( \gamma \) is a pdp in \( D \) starting at \( {P}_{1} \) and ending at \( {P}_{2} \), then | \[ {\int }_{\gamma }\mathrm{d}F = F\left( {P}_{2}\right) - F\left( {P}_{1}\right) \] | Yes |
Theorem 4.16. Let \( \omega \) be a differential form on an open disc \( U \) . Then \( \omega \) is exact on \( U \) if and only if \( {\int }_{\gamma }\omega = 0 \) for all \( \gamma \) that are boundaries of rectangles contained in \( U \) with sides parallel to the coordinate axes. | Proof. Repeat the appropriate argument in the proof of the last lemma with \( \left( {{x}_{0},{y}_{0}}\right) \) the center of \( U \), observing that any other point in the disc may be joined to the center by either a vertical segment, a horizontal segment, or two consecutive segments, one horizontal and one vertical. | No |
Corollary 4.17. A differential form \( \omega \) is closed on an open disc if and only if it is exact on the disc. | Proof. Every exact form on a domain is closed, so we just need to show that the converse holds on an open disc. So assume \( \omega \) is a closed form on the open disc \( U \) . By the theorem, it is enough to show that if \( R \) is any rectangle contained in \( U \) , with sides parallel to the coordinate axes, then... | Yes |
Theorem 4.20 (Green’s Theorem). Let \( R \) be an \( \left( {xy}\right) \) -simple region and let \( \gamma \) denote its boundary oriented counterclockwise (this means that \( R \) lies to the left of the oriented curves on its boundary). Consider a \( {\mathbf{C}}^{1} \) -form \( \omega = P\mathrm{\;d}x + Q\mathrm{\;... | Proof. Using the notation introduced in the definition of \( \left( {xy}\right) \) -simple regions, we have\n\n\[ \n{\iint }_{R}\frac{\partial Q}{\partial x}\mathrm{\;d}x\mathrm{\;d}y = {\int }_{c}^{d}{\int }_{{h}_{1}\left( y\right) }^{{h}_{2}\left( y\right) }\frac{\partial Q}{\partial x}\mathrm{\;d}x\mathrm{\;d}y \n\]... | Yes |
Theorem 4.23. Suppose that \( \omega = P\mathrm{\;d}x + Q\mathrm{\;d}y \) is a \( {\mathbf{C}}^{1} \) -differential form on a domain D. If \( \omega \) is closed, then \( {P}_{x} = {Q}_{y} \). Conversely, if \( D \) is an open disc, \( P \) and \( Q \) are \( {\mathbf{C}}^{1} \) -functions on \( D \), and \( {P}_{x} = ... | Proof. If \( \omega \) is closed in the domain \( D \), then near every point in \( D \) there exists a function \( F \) such that \( \omega = \mathrm{d}F = {F}_{x}\mathrm{\;d}x + {F}_{y}\mathrm{\;d}y \) . But \( \omega \) is \( {\mathbf{C}}^{1} \) and thus \( F \) is \( {\mathbf{C}}^{2} \) ; therefore \( {P}_{y} = {F}... | Yes |
Corollary 4.24. If \( \omega = P\mathrm{\;d}x + Q\mathrm{\;d}y \) is a \( {\mathbf{C}}^{1} \) -form on a domain \( D \), then \( \omega \) is closed on \( D \) if and only if \( {P}_{y} = {Q}_{x} \) in \( D \) . | Proof. For any point in \( D \), consider an open disc \( U \) centered at that point and contained in \( D \), and apply the previous theorem to \( \omega \) restricted to \( U \) . | No |
Lemma 4.26. Let \( f\left( z\right) \mathrm{d}z \) be of class \( {\mathbf{C}}^{1} \) on a domain \( D \) . Then \( f\left( z\right) \mathrm{d}z \) is closed on \( D \) if and only if \( f \) is holomorphic in \( D \) . | Proof. By the above remarks and previous Corollary, \( f\left( z\right) \mathrm{d}z \) is a closed form on \( D \) if and only if \( {u}_{y} = - {v}_{x} \) and \( {v}_{y} = {u}_{x} \) if and only if \( u \) and \( v \) satisfy CR if and only if \( f \) is holomorphic. | Yes |
Lemma 4.27. A \( {\mathbf{C}}^{1} \) -function \( F \) is a primitive for \( f\left( z\right) \mathrm{d}z \) if and only if \( {F}^{\prime } = f \) . | Proof. The function \( F \) is a primitive for \( f\left( z\right) \mathrm{d}z \) if and only if \( \mathrm{d}F = {F}_{z}\mathrm{\;d}z + {F}_{\bar{z}}\mathrm{\;d}\bar{z} = \) \( f\left( z\right) \mathrm{d}z \) if and only if \( {F}_{\bar{z}} = 0 \) and \( {F}_{z} = {F}^{\prime } = f \) . | Yes |
Not every closed form is exact. Let \( D = {\mathbb{C}}_{ \neq 0} \) and \( \omega = \frac{\mathrm{d}z}{z} \) . | (a) If \( \gamma \left( t\right) = {\mathrm{e}}^{2\pi \iota t} \) for \( t \in \left\lbrack {0,1}\right\rbrack \), then \( {\int }_{\gamma }\omega = {2\pi \iota } \) . Thus \( \omega \) is not exact on \( D \) .\n\n(b) Since \( f\left( z\right) = \frac{1}{z} \) is holomorphic and \( {\mathbf{C}}^{1} \) on \( D,\omega \... | Yes |
We use Theorem 4.32 to compute \( {\int }_{\gamma }\omega \), where \( \omega \) is a closed differential form in \( D \) and \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow D \) is a pdp in \( {D}^{\prime } \) . | Subdivide \( \left\lbrack {a, b}\right\rbrack = {I}_{0} \cup {I}_{1} \cup \cdots \cup {I}_{n} \), where \( {I}_{j} = \left\lbrack {{a}_{j},{a}_{j + 1}}\right\rbrack ,{a}_{0} = a \), and \( {a}_{n + 1} = b \), such that \( {\left. {\gamma }_{j} = \gamma \right| }_{{I}_{j}} \) is a differentiable path and \( \omega \) ha... | Yes |
For every \( c \in \mathbb{C} \) and every continuous closed path \( \gamma \) in \( \mathbb{C} - \{ c\} \) , the number\n\n\[ \n\frac{1}{{2\pi }\imath }{\int }_{\gamma }\frac{\mathrm{d}z}{z - c} \in \mathbb{Z} \n\] | Proof. We may assume \( c = 0 \) . Let \( f \) be a primitive of \( \frac{\mathrm{d}z}{z} \) along the curve \( \gamma \) . Then\n\n\[ \n{\int }_{\gamma }\frac{\mathrm{d}z}{z} = f\left( b\right) - f\left( a\right) \n\]\n\nwhere \( \left\lbrack {a, b}\right\rbrack \) parameterizes \( \gamma \) . Since \( \gamma \left( a... | Yes |
Example 4.39 (In polar coordinates). Let \( r = g\left( \theta \right) > 0 \), with \( g \in {\mathbf{C}}^{1}\left( \mathbb{R}\right) \) . Let \( n \in \) \( {\mathbb{Z}}_{ > 0} \) and define \( \gamma \left( \theta \right) = g\left( \theta \right) {\mathrm{e}}^{\iota \theta } \), where \( \theta \in \left\lbrack {0,{2... | \[ I\left( {\gamma ,0}\right) = \frac{1}{{2\pi }\imath }{\int }_{\gamma }\frac{\mathrm{d}z}{z} = \frac{1}{{2\pi }\imath }{\int }_{0}^{2\pi n}\frac{d\left( {g\left( \theta \right) {\mathrm{e}}^{\imath \theta }}\right) }{g\left( \theta \right) {\mathrm{e}}^{\imath \theta }} \] \[ = \frac{1}{{2\pi }\imath }{\int }_{0}^{2\... | Yes |
Theorem 4.45. If \( \omega \) is a closed form on \( D \) and \( \delta : \left\lbrack {0,1}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \rightarrow D \) is a continuous map, then a primitive \( f \) for \( \omega \) along \( \delta \) exists and is unique up to an additive constant. | Proof. We leave the proof as an exercise for the reader. | No |
Theorem 4.46. Let \( {\gamma }_{0} \) and \( {\gamma }_{1} \) be continuous paths in a domain \( D \) and let \( \omega \) be a closed form on \( D \) . If \( {\gamma }_{0} \) is homotopic to \( {\gamma }_{1} \) with fixed end points, then\n\n\[{\int }_{{\gamma }_{0}}\omega = {\int }_{{\gamma }_{1}}\omega\] | Proof. We assume that both paths are parameterized by the interval \( I = \left\lbrack {0,1}\right\rbrack \) . Let \( \delta : I \times I \rightarrow D \) be a homotopy between our two paths and let \( f \) be a primitive of \( \omega \) along \( \delta \) . Thus \( u \mapsto f\left( {0, u}\right) \) is a primitive of ... | Yes |
Corollary 4.54. In every simply connected domain not containing the point 0, there exists a branch of \( \log z \) . | Proof. The differential form \( \omega = \frac{\mathrm{d}z}{z} \) is closed and thus exact in the given domain. Hence there exists a holomorphic function \( F \) (on the same domain) such that \( \mathrm{d}F = \) \( \omega \) . This function \( F \) is a branch of the logarithm. | Yes |
Theorem 4.57. Let \( f : \{ z \in \mathbb{C} : \left| z\right| \leq R\} \rightarrow \mathbb{C} \) be a continuous map (with \( R > 0 \) ) and let \( \gamma \left( \theta \right) = f\left( {R{\mathrm{e}}^{{2\pi }\imath \theta }}\right) \) for \( \theta \in \left\lbrack {0,1}\right\rbrack \) . If \( c \notin \) range \( ... | Proof. Assume \( f\left( z\right) \neq c \) for all \( \left| z\right| < R \) . Then \( f\left( z\right) \neq c \) for all \( \left| z\right| \leq R \), because \( c \notin \) range \( \gamma \) .\n\nDefine \( \delta \left( {\rho ,\theta }\right) = f\left( {{\rho R}{\mathrm{e}}^{{2\pi }\imath \theta }}\right) \) on \( ... | Yes |
Theorem 4.59. If \( {\gamma }_{1} \) and \( {\gamma }_{2} \) are continuous closed paths not passing through 0, then\n\n\[ I\left( {{\gamma }_{1}{\gamma }_{2},0}\right) = I\left( {{\gamma }_{1},0}\right) + I\left( {{\gamma }_{2},0}\right) . \] | Proof. Let \( \omega = \frac{\mathrm{d}z}{z} \) and \( {\gamma }_{j} : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{C} - \{ 0\} \) . Choose continuous functions \( {f}_{j} \) : \( \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{C} \) so that \( {\mathrm{e}}^{{f}_{j}\left( t\right) } = {\gamma }_{j}\left( t\r... | Yes |
Theorem 4.60. Let \( {\gamma }_{1} \) and \( \gamma \) be continuous closed paths in \( \mathbb{C} \) parameterized by \( \left\lbrack {0,1}\right\rbrack \) . Assume that\n\n\[ 0 < \left| {{\gamma }_{1}\left( t\right) }\right| < \left| {\gamma \left( t\right) }\right| \text{ for all }t \in \left\lbrack {0,1}\right\rbra... | Proof. Note that\n\n\[ \gamma \left( t\right) + {\gamma }_{1}\left( t\right) = \gamma \left( t\right) \left( {1 + \frac{{\gamma }_{1}\left( t\right) }{\gamma \left( t\right) }}\right) = \gamma \left( t\right) \beta \left( t\right) \]\n\nwith \( \beta \left( t\right) = 1 + \frac{{\gamma }_{1}\left( t\right) }{\gamma \le... | Yes |
Theorem 5.1 (Goursat’s Theorem, Strengthened Version). If \( f \) is continuous in a domain \( D \) and holomorphic except possibly on a line segment in \( D \), then \( f\left( z\right) \mathrm{d}z \) is closed in \( D \) . | Proof. Without loss of generality, \( D \) is the unit disc, and the line segment is all or part of the real axis in \( D \) .\n\nWe must show that the integral \( {\int }_{\gamma }f\left( z\right) \mathrm{d}z \) vanishes whenever \( \gamma \) is the (positively oriented) boundary of an open rectangle \( R \) whose clo... | Yes |
Theorem 5.2 (Cauchy’s Integral Formula). If \( f \) is holomorphic on a domain \( D \) and \( \gamma \) is a continuous closed path homotopic to a point in \( D \), then for all \( c \in \) \( \mathbb{C} \) - range \( \gamma \), we have\n\n\[ \frac{1}{{2\pi }\imath }{\int }_{\gamma }\frac{f\left( z\right) }{z - c}\math... | Proof. Define, for \( z \in D \) ,\n\n\[ g\left( z\right) = \left\{ \begin{array}{ll} \frac{f\left( z\right) - f\left( c\right) }{z - c}, & \text{ if }z \neq c, \\ {f}^{\prime }\left( c\right) , & \text{ if }z = c. \end{array}\right. \]\n\nThen \( g \) is continuous on \( D \) and holomorphic except (possibly) at \( c ... | Yes |
Let \( D \) be a domain in \( \mathbb{C}, f \) a holomorphic function defined on \( D \) , and \( c \in D \) . Choose \( R > 0 \) such that \( \operatorname{cl}U\left( {c, R}\right) \subset D \) and let \( \gamma \left( \theta \right) = c + R{\mathrm{e}}^{{2\pi }\imath \theta } \), for \( 0 \leq \theta \leq 1 \) . Then... | \[ \frac{1}{{2\pi }\imath }{\int }_{\gamma }\frac{f\left( z\right) }{z - w}\mathrm{\;d}z = f\left( w\right) \text{ for }\left| {w - c}\right| < R\text{ and } \] \[ \frac{1}{{2\pi }\imath }{\int }_{\gamma }\frac{f\left( z\right) }{z - w}\mathrm{\;d}z = 0\text{ for }\left| {w - c}\right| > R. \] | Yes |
Corollary 5.6. A function \( f \) is holomorphic in an open set \( D \) if and only if \( f \) has a power series expansion at each point of \( D \) . For a holomorphic function \( f \) on \( D \) , the Taylor series expansion of \( f \) at \( c \in D \) | \[ f\left( z\right) = \mathop{\sum }\limits_{{k = 0}}^{\infty }{a}_{k}{\left( z - c\right) }^{k} \] has radius of convergence \[ \rho \geq \sup \{ r > 0;U\left( {c, r}\right) \subseteq D\} . \] | Yes |
Corollary 5.9 (Cauchy’s Generalized Integral Formula). Let \( f \) be holomorphic on a domain \( D \) containing \( \operatorname{cl}U\left( {c, R}\right) \) for some \( c \in D \) and \( R > 0 \) . If \( \gamma \left( \theta \right) = \) \( c + R{\mathrm{e}}^{\iota \theta } \) for \( 0 \leq \theta \leq {2\pi } \), the... | Proof. Recall that for \( n = 0,1,2,\ldots \) , \[ {a}_{n} = \frac{1}{{2\pi }\imath }{\int }_{\gamma }\frac{f\left( t\right) }{{\left( t - c\right) }^{n + 1}}\mathrm{\;d}t = \frac{{f}^{\left( n\right) }\left( c\right) }{n!}. \] | Yes |
Theorem 5.10 (Morera’s Theorem). If \( f \in {\mathbf{C}}^{0}\left( D\right) \) and \( f\left( z\right) \mathrm{d}z \) is closed on \( D \) , then \( f \) is holomorphic on \( D \) . | Proof. Since the differential form \( \omega = f\left( z\right) \mathrm{d}z \) is locally exact, for each point \( c \in D \) there is a neighborhood \( U \) of \( c \) in \( D \) and a primitive \( F \) of \( \omega \) in \( U \) . That is, there is a \( {\mathbf{C}}^{1} \) -function \( F \) on \( U \) with \( {F}_{z}... | Yes |
Theorem 5.16 (Fundamental Theorem of Algebra). If \( P \) is a polynomial of degree \( n \geq 1 \), then there exist \( {a}_{1},\ldots ,{a}_{n} \in \mathbb{C} \) and \( b \in {\mathbb{C}}_{ \neq 0} \) such that\n\n\[ P\left( z\right) = b\mathop{\prod }\limits_{{j = 1}}^{n}\left( {z - {a}_{j}}\right) \text{ for all }z \... | Proof. It suffices to show that \( P \) has a root. If not, \( \frac{1}{P} \) is an entire function. It is also bounded since \( \mathop{\lim }\limits_{{z \rightarrow \infty }}\frac{1}{P\left( z\right) } = 0 \) and thus must be constant. | Yes |
Theorem 5.26 (Cauchy’s Theorem (Extended Version)). Let \( {\gamma }_{0},\ldots ,{\gamma }_{n} \) be \( n + 1 \) positively oriented Jordan curves. Assume that\n\n\[ \text{range}{\gamma }_{j} \subset e\left( {\gamma }_{k}\right) \cap i\left( {\gamma }_{0}\right) \]\n\nfor all \( 1 \leq j \neq k \leq n \), see Fig. 5.2.... | Proof. Adjoin nonintersecting curves \( {\delta }_{j} \) in \( D \) from \( {\gamma }_{0} \) to \( {\gamma }_{j} \) for \( j = 1,\ldots, n \), as in Fig. 5.2. Then the cycle\n\n\[ \delta = \left( {{\gamma }_{0},{\delta }_{1} * {\gamma }_{{1}_{ - }} * {\delta }_{{1}_{ - }},\ldots ,{\delta }_{n} * {\gamma }_{{n}_{ - }} *... | Yes |
Theorem 5.27 (Cauchy's Integral Formula (Extended Version)). With the hypotheses as in the extended version of Cauchy's Theorem 5.26, we have\n\n\[ \n{2\pi }\imath f\left( c\right) = {\int }_{{\gamma }_{0}}\frac{f\left( z\right) }{z - c}\mathrm{\;d}z - \mathop{\sum }\limits_{{k = 1}}^{n}{\int }_{{\gamma }_{k}}\frac{f\l... | Proof. We can apply Theorem 5.2 to the function \( f \), using the neighborhood \( N \) of Theorem 5.26 and the cycle \( \delta \) constructed in its proof, since \( \delta \) is homologous to zero in \( N \) and \( I\left( {\delta, c}\right) = + 1 \) . As before, the integral over each \( {\delta }_{k} \) is canceled ... | Yes |
Theorem 5.30 (Maximum Modulus Principle). Suppose \( f \) is a continuous complex-valued function defined on a domain \( D \) in \( \mathbb{C} \) that has the MVP. If \( \left| f\right| \) has a relative maximum at a point \( c \in D \), then \( f \) is constant in a neighborhood of \( c \) . | Proof. The result is clear if \( f\left( c\right) = 0 \) . If \( f\left( c\right) \neq 0 \), replacing \( f \) by \( {\mathrm{e}}^{-{t\theta }}f \) for some \( \theta \in \mathbb{R} \), we may assume that \( f\left( c\right) > 0 \) . Write \( f = u + {\iota v} \) and choose \( {r}_{0} > 0 \) such that\n\n(1) \( \operat... | Yes |
Corollary 5.31. Suppose \( D \) is a bounded domain and \( f \in {\mathbf{C}}^{0}\left( {\mathrm{{cl}}D}\right) \) satisfies the MVP in D. If\n\n\[ M = \sup \{ \left| {f\left( z\right) }\right| ;z \in \partial D\} \]\n\nthen\n\n(a) \( \left| {f\left( z\right) }\right| \leq M \) for all \( z \in D \) .\n\n(b) If \( \lef... | Proof. If\n\n\[ {M}^{\prime } = \sup \{ \left| {f\left( z\right) }\right| ;z \in \operatorname{cl}D\} ,\]\n\nthen\n\n\[ M \leq {M}^{\prime } < + \infty \]\n\nWe know that there exists a \( c \) in \( \operatorname{cl}D \) such that \( \left| {f\left( c\right) }\right| = {M}^{\prime } \) . If \( c \in D \), then \( f \)... | Yes |
Theorem 5.34 (Schwarz’s Lemma). If \( f \) is a holomorphic function defined on \( U\left( {0,1}\right) \) satisfying \( \left| {f\left( z\right) }\right| < 1 \) for \( \left| z\right| < 1 \) and \( f\left( 0\right) = 0 \), then \( \left| {f\left( z\right) }\right| \leq \left| z\right| \) for \( \left| z\right| < 1 \) ... | Proof. Using the Taylor series expansion for \( f \) at 0, we can write \( f\left( z\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{z}^{n} \);\n\nthis power series has radius of convergence \( \rho \geq 1 \), by Theorem 5.5. Then the function defined by\n\n\[ g\left( z\right) = \left\{ \begin{array}{l} \frac... | Yes |
Theorem 5.37 (Cauchy’s Integral Formula for Smooth Functions). Let \( K \) be a compact set in \( \mathbb{C} \) that is the closure of its interior, with piecewise smooth positively oriented boundary \( \partial K \) . If \( f \) is a \( {\mathbf{C}}^{1} \) -function on a neighborhood of \( K \) and \( c \) is a point ... | Proof. Choose \( \epsilon > 0 \) such that the closure of the ball \( U\left( {c,\epsilon }\right) \) is contained in the interior of \( K \), and let \( {K}_{\epsilon } = K - U\left( {c,\epsilon }\right) \) . We apply Green’s theorem to the smooth differential form \( \frac{f\left( z\right) }{z - c}\mathrm{\;d}z \) on... | Yes |
Theorem 6.1 (Laurent Series Expansion). Let \( c \in \mathbb{C} \) and let \( f \) be holomorphic in the annulus\n\n\[ A = \\left\\{ {z \in \mathbb{C};0 \leq {R}_{1} < \\left| {z - c}\\right| < {R}_{2} \leq + \infty }\\right\\} .{}^{1} \]\n\nThen\n\n\[ f\\left( z\\right) = \\mathop{\\sum }\\limits_{{n = - \infty }}^{\\... | Proof. Without loss of generality we assume \( c = 0 \) . Consider two concentric circles \( {\\gamma }_{{r}_{j}} = \\left\\{ {z;\\left| z\\right| = {r}_{j}}\\right\\} \\left( {j = 1,2}\\right) \), bounding a smaller annulus\n\n\[ {R}_{1} < {r}_{1} < \\left| z\\right| < {r}_{2} < {R}_{2} \]\n\nIf for \( j \in \\{ 1,2\\... | Yes |
Corollary 6.2. If \( f \) is holomorphic in \( A \), then \( f = {f}_{2} - {f}_{1} \) where \( {f}_{2} \) is holomorphic in \( \left| {z - c}\right| < {R}_{2} \) and \( {f}_{1} \) is holomorphic in \( \left| {z - c}\right| > {R}_{1} \) (including the point at \( \infty \) ). The functions \( {f}_{j} \) are unique if we... | Proof. Once again, without loss of generality, we assume \( c = 0 \ | No |
The rational function\n\n\[ f\\left( z\\right) = \\frac{1}{z\\left( {1 - z}\\right) \\left( {2 - z}\\right) }\n\]\n\nis holomorphic on three annuli centered at zero. On each of these it has a different Laurent series expansion of the form \( \\mathop{\\sum }\\limits_{{n = - \\infty }}^{{+\\infty }}{a}_{n}{z}^{n} \) ; t... | (1) On \( {A}_{1} = \\{ 0 < \\left| z\\right| < 1\\} \) : we do not need to evaluate the coefficients \( {a}_{n} \) by the integration process described in the proof of the last theorem. On this annulus, we have\n\n\[ \\frac{1}{1 - z} = 1 + z + {z}^{2} + \\cdots + {z}^{n} + \\cdots \\text{ (valid for }\\left| z\\right|... | Yes |
Theorem 6.7. Let \( c \in \mathbb{C} \) and \( f \) be a holomorphic function on \( A = \{ 0 < \left| {z - c}\right| < \) \( \left. {R}_{2}\right\} \) . Assume that (6.1) is the Laurent series of \( f \) on the punctured disc A. If there exist an \( M > 0 \) and \( 0 < {r}_{0} < {R}_{2} \) such that\n\n\[ \left| {f\lef... | Proof. We know that \( {a}_{n} = \frac{1}{{2\pi }\imath }{\int }_{{\gamma }_{r}}\frac{f\left( t\right) }{{\left( t - c\right) }^{n + 1}}\mathrm{\;d}t \) for all \( n \in \mathbb{Z} \), where \( {\gamma }_{r}\left( \theta \right) = \) \( c + r{\mathrm{e}}^{\iota \theta } \), for \( \theta \in \left\lbrack {0,{2\pi }}\ri... | Yes |
Theorem 6.8 (Casorati-Weierstrass). If \( f \) is holomorphic on \( \{ 0 < \left| {z - c}\right| < \) \( \left. {R}_{2}\right\} \) and has an essential singularity at \( z = c \), then for all \( w \in \mathbb{C} \) the function\n\n\[ g\left( z\right) = \frac{1}{f\left( z\right) - w} \]\n\nis unbounded in any punctured... | Proof. Assume that, for some \( w \in \mathbb{C} \), the function \( g \) is unbounded in some punctured neighborhood \( N \) of \( z = c \) . Then there is \( \varepsilon > 0 \) such that \( N = \) \( U\left( {c,\varepsilon }\right) - \{ c\} \), and, for any \( M > 0 \), there exists a \( z \in N \) such that \( \left... | Yes |
The function \( \exp \left( \frac{1}{z}\right) \) shows the above theorem is sharp, with \( c = \) 0 and \( {w}_{0} = 0 \). | For a proof, see Conway’s book listed in the bibliography. | No |
Theorem 6.11. Assume that \( f \) is a holomorphic function in a punctured disc \( {U}^{\prime } = \) \( U\left( {c, R}\right) - \{ c\} \) around the isolated singularity \( c \in \mathbb{C} \) . Then\n\n(1) \( c \) is a removable singularity if and only if \( f \) is bounded in \( {U}^{\prime } \) if and only if \( \m... | Proof. (1) follows from Theorem 6.7 and the definition of removable singularity. | No |
Theorem 6.13. Let \( f : \mathbb{C} \cup \{ \infty \} \rightarrow \mathbb{C} \cup \{ \infty \} \) . Then,\n\n(a) If \( f \) is holomorphic, it is constant, and\n\n(b) If \( f \) is meromorphic, it is a rational function. | Proof. If \( f \) is holomorphic on \( \mathbb{C} \cup \{ \infty \} \), a compact set, it must be bounded. Since it is also an entire function, it must be constant, by Liouville's theorem.\n\nIf \( f \) is meromorphic on \( \mathbb{C} \cup \{ \infty \} \), its set of poles must be finite, being isolated points in a com... | Yes |
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