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Corollary 48.2.15 Let \( X \) be a Banach space and let \( K \) be a compact convex subset. Let \( f : K \times \Omega \rightarrow K \) satisfy\n\n\[ x \rightarrow f\left( {x,\omega }\right) \text{is continous} \]\n\n\[ \omega \rightarrow f\left( {x,\omega }\right) \text{is measurable} \]\n\nThen \( f\left( {\cdot ,\om... | Proof: The set \( K \) has a countable dense subset \( \left\{ {k}_{i}\right\} \) . You could consider \( Y \) as the closure in \( X \) of the span of these \( {k}_{i} \) . Thus \( Y \) is a separable Banach space which contains \( K \) . Now apply the above result. - | No |
Theorem 48.2.16 Let \( f\left( {\cdot ,\omega }\right) : X \rightarrow X \) be a compact map (takes bounded sets to precompact sets and continuous) where \( X \) is a Banach space. Also suppose that\n\n\[ \sup \{ z \in f\left( {B\left( {0, r}\right) ,\omega }\right) \} \leq C\left( r\right) \]\n\nindependent of \( \ome... | Proof: Suppose that alternative 2 does not hold and yet alternative 1 also fails to hold. Since alternative 2 does not hold, there exists \( {M}_{0} \) such that for all \( \omega \), and for all \( t \in \left( {0,1}\right) \), if \( x = {tf}\left( {x,\omega }\right) \), then \( \parallel x\left( \omega \right) \paral... | Yes |
Lemma 48.2.17 Let \( X \equiv C\left( {\left\lbrack {0, T}\right\rbrack ;{\mathbb{R}}^{n}}\right) \) and let \( \mathbf{f}\left( {\cdot ,\cdot ,\omega }\right) : \left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) where \( \omega \in \Omega \) for \( \left( {\Omega ,\mathcal{F}}\rig... | Proof: The space \( X \) is separable and so by the Riesz representation theorem and the Pettis theorem, it suffices to verify that for \( \mathbf{y} \in X \), \n\n\[ \omega \rightarrow {\int }_{0}^{T}{\int }_{0}^{t}\mathbf{f}\left( {s,\mathbf{y}\left( s\right) + {\mathbf{x}}_{0}\left( \omega \right) ,\omega }\right) {... | Yes |
Theorem 48.2.18 Let \( \mathbf{f}\left( {\cdot ,\cdot ,\omega }\right) : \left\lbrack {0, T}\right\rbrack \times {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) be continuous and suppose\n\n\[ \omega \rightarrow F\left( {\mathbf{y},\omega }\right) \]\n\n\( \left( {48.2.6}\right) \)\n\nis measurable into \( C\left( {\l... | Proof: Let \( F\left( {\cdot ,\omega }\right) : X \rightarrow X \) where \( X \) described above.\n\n\[ F\left( {\mathbf{y},\omega }\right) \left( t\right) \equiv {\int }_{0}^{t}\mathbf{f}\left( {s,\mathbf{y}\left( s\right) + {\mathbf{x}}_{0},\omega }\right) {ds} \]\n\n\nF is clearly continuous in the first variable an... | Yes |
Theorem 48.3.1 Let \( V \) be a reflexive separable Banach space. Suppose \( \omega \rightarrow \) \( A\left( {u,\omega }\right) \) has a measurable selection in \( {V}^{\prime } \), for each \( u \in V \) and \( \omega \in \Omega \) the set \( A\left( {u,\omega }\right) \) is closed and convex in \( {V}^{\prime } \) a... | Proof: Let \( \omega \rightarrow u\left( \omega \right) \) be measurable into \( V \), and let \( {u}_{n}\left( \omega \right) \rightarrow u\left( \omega \right) \) in \( V \) where \( {u}_{n} \) is a simple function\n\n\[ \n{u}_{n}\left( \omega \right) = \mathop{\sum }\limits_{{k = 1}}^{{m}_{n}}{c}_{k}^{n}{\mathcal{X}... | Yes |
Theorem 48.3.3 Assumtions: Let \( B\left( {\cdot ,\omega }\right) : V \rightarrow \mathcal{P}\left( {V}^{\prime }\right), C\left( {\cdot ,\omega }\right) : V \rightarrow \mathcal{P}\left( {V}^{\prime }\right) \) for \( V \) a separable Banach space. Suppose that \( \omega \rightarrow B\left( {x,\omega }\right) ,\omega ... | Proof: The argument will refer to the following commutative diagram.\n\n\[ \begin{matrix} E{\left( \omega \right) }^{\prime } & \overset{\theta {\left( \omega \right) }^{ * }}{ \rightarrow } & {\mathbb{R}}^{n} \\ i{\left( \omega \right) }^{ * }A\left( {\cdot ,\omega }\right) \uparrow & & \uparrow \theta {\left( \omega ... | Yes |
Theorem 48.5.3 Let \( V \) be a reflexive separable Banach space. Let \( \omega \rightarrow K\left( \omega \right) \) be a measurable multifunction, \( K\left( \omega \right) \) convex, closed, and bounded. Also for \( A = B, C \) let \( A\left( {\cdot , \cdot }\right) \) satisfy 1 - 3. Let \( \omega \rightarrow f\left... | Proof: Let \( {V}_{n} = {V}_{n}\left( \omega \right) \) denote an increasing sequence of finite dimensional subspaces whose union is dense in \( V \) . Let \( {V}_{n}\left( \omega \right) \) contain the first \( n \) vectors of \( {\left\{ {d}_{k}\left( \omega \right) \right\} }_{k = 1}^{\infty } \) where the closure o... | Yes |
We can let \( \Omega = \left\lbrack {0, T}\right\rbrack \) and let the measurable sets be the Lebesgue measurable sets, \( t \rightarrow f\left( t\right) \) measurable into \( {V}^{\prime } \) . Then for \( A\left( {\cdot , \cdot }\right) \) satisfying 1 - 3 the above theorem gives the solution \( u, w\left( t\right), ... | If \( u \rightarrow A\left( {u, t}\right) \) is coercive, this allows for \( K\left( t\right) \) only closed and convex. If \( A \) is the sum of \( B, C \) and \( u \rightarrow B\left( {u, t}\right) \) and \( u \rightarrow C\left( {u, t}\right) \), these each satisfying the conditions 1 - 3, \( w\left( t\right) = {w}_... | Yes |
Lemma 48.6.1 Let \( F \) be those points where \( u = 0 \) . Then for a.e. \( \mathbf{x} \in F,\nabla u\left( \mathbf{x}\right) = \) 0. Here \( u \in {W}^{1, p}\left( \Omega \right) \) and we assume \( \partial \Omega \) has measure zero. | Proof: It suffices to consider \( u = 0 \) on \( F \) contained in the interior of \( \Omega \) . First I show that \( {u}_{,{x}_{n}} = 0 \) a.e.\n\n\[ \frac{u\left( {{t}_{1},\cdots ,{t}_{n - 1},{t}_{n}}\right) - u\left( {{t}_{1},\cdots ,{t}_{n - 1},{t}_{n} - h}\right) }{h} = \frac{1}{h}{\int }_{{t}_{n} - h}^{{t}_{n}}{... | Yes |
Lemma 48.6.2 Let \( V \) be a closed subset of \( {W}^{1, p}\left( \Omega \right), p > 1 \) and let \( k \in V \) . Then \( \max \left( {k, u}\right) \in V \) and if \( {u}_{n} \rightarrow u \) in \( V \), then \( \max \left( {{u}_{n}, k}\right) \rightarrow \max \left( {u, k}\right) \) in \( V \) . | Proof: We consider \( \psi \left( r\right) = \left| r\right| ,{\psi }_{\varepsilon }\left( r\right) = \sqrt{\varepsilon + {r}^{2}} \) . Then for \( \phi \in {C}_{c}^{\infty }\left( \Omega \right) \) ,\n\n\[{\int }_{\Omega }\psi \left( {u\left( \mathbf{x}\right) }\right) \phi ,{x}_{k}\left( \mathbf{x}\right) = \mathop{\... | Yes |
Theorem 48.7.3 Suppose conditions 1 - 5 hold. Also, suppose\n\n\[ V \subseteq H = {H}^{\prime } \subseteq {V}^{\prime }\text{, where}V\text{is dense in}H\text{,}\]\n\nThen, the operator \( \widehat{A} \) satisfies the following.\n\nHypotheses:\n\n\[ {u}_{n} \rightarrow u\text{ weakly in }\mathcal{V},\lim \mathop{\sup }... | Proof: It was argued above that \( \widehat{A}\left( u\right) \) is closed and convex.\n\nEnlarge the set of measure zero \( \sum \), if needed, so that for each \( n \),\n\n\[ {z}_{n}\left( t\right) \in A\left( {{u}_{n}\left( t\right), t}\right)\]\n\nfor each \( t \notin \sum \).\n\nNext, we claim that if \( t \notin ... | Yes |
Theorem 48.7.4 Suppose conditions 1 - 5 hold. Also, suppose \( U \) is a separable Hilbert space dense in \( V \), a reflexive separable Banach space with the inclusion map compact and \( V \) is dense in a Hilbert space \( H \) . Thus\n\n\[ U \subseteq V \subseteq H = {H}^{\prime } \subseteq {V}^{\prime } \subseteq {U... | Proof: It was argued above that \( \widehat{A}\left( u\right) \) is closed and convex. Let \( \sum \) have measure zero and for each \( \left( {t,\omega }\right) \notin \sum ,{z}_{n}\left( {t,\omega }\right) \in A\left( {{u}_{n}\left( {t,\omega }\right), t,\omega }\right) \) for each \( n \) . Now \( {\sum }_{\omega } ... | Yes |
Proposition 49.0.2 The complex numbers with the norm just mentioned forms a complete normed linear space. | Proof: Let \( \left\{ {z}_{n}\right\} \) be a Cauchy sequence of complex numbers with \( {z}_{n} = {x}_{n} + i{y}_{n} \) . Then \( \left\{ {x}_{n}\right\} \) and \( \left\{ {y}_{n}\right\} \) are Cauchy sequences of real numbers and so they converge to real numbers, \( x \) and \( y \) respectively. Thus \( {z}_{n} = {... | Yes |
Theorem 49.0.6 Let \( \\left\\{ {f}_{n}\\right\\} \) be a sequence of complex valued functions defined on \( S \\subseteq \\mathbb{C} \) . Suppose there exists \( {M}_{n} \) such that \( {\\begin{Vmatrix}{f}_{n}\\end{Vmatrix}}_{\\infty } < {M}_{n} \) and \( \\sum {M}_{n} \) converges. Then \( \\sum {f}_{n} \) converges... | Proof: Let \( z \\in S \) . Then letting \( m < n \)\n\n\[\\begin{Vmatrix}{\\mathop{\\sum }\\limits_{{k = 1}}^{n}{f}_{k}\\left( z\\right) - \\mathop{\\sum }\\limits_{{k = 1}}^{m}{f}_{k}\\left( z\\right) }\\end{Vmatrix} \\leq \\mathop{\\sum }\\limits_{{k = m + 1}}^{n}\\begin{Vmatrix}{{f}_{k}\\left( z\\right) }\\end{Vmat... | Yes |
Theorem 49.1.2 \( \left( {\widehat{\mathbb{C}}, d}\right) \) is a compact, hence complete metric space. | Proof: Suppose \( \left\{ {z}_{n}\right\} \) is a sequence in \( \widehat{\mathbb{C}} \) . This means \( \left\{ {\theta \left( {z}_{n}\right) }\right\} \) is a sequence in \( {S}^{2} \) which is compact. Therefore, there exists a subsequence, \( \left\{ {\theta {z}_{{n}_{k}}}\right\} \) and a point, \( z \in {S}^{2} \... | Yes |
Theorem 50.0.3 Let \( \phi \) and \( \gamma \) be as just described. Then assuming that\n\n\[{\int }_{\gamma }{fd\gamma }\]\n\nexists, so does\n\n\[{\int }_{\gamma \circ \phi }{fd}\left( {\gamma \circ \phi }\right)\]\n\nand\n\n\[{\int }_{\gamma }{fd\gamma } = {\int }_{\gamma \circ \phi }{fd}\left( {\gamma \circ \phi }\... | Proof: There exists \( \delta > 0 \) such that if \( \mathcal{P} \) is a partition of \( \left\lbrack {a, b}\right\rbrack \) such that \( \parallel \mathcal{P}\parallel < \delta \) ,\nthen\n\n\[ \left| {{\int }_{\gamma }{fd\gamma } - S\left( \mathcal{P}\right) }\right| < \varepsilon \]\n\nBy continuity of \( \phi \), t... | Yes |
Theorem 50.0.5 Let \( f \in C\left( {\gamma }^{ * }\right) \) and let \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{C} \) be of bounded variation and continuous. Let \[ M \geq \max \{ \parallel f \circ \gamma \left( t\right) \parallel : t \in \left\lbrack {a, b}\right\rbrack \} . \] Then \[ \left| \l... | Proof: Let 50.0.5 hold. From the proof of the above theorem, when \( \parallel \mathcal{P}\parallel < {\delta }_{m} \) , \[ \left| \left| {{\int }_{\gamma }{fd\gamma } - S\left( \mathcal{P}\right) }\right| \right| \leq \frac{2}{m}V\left( {\gamma ,\left\lbrack {a, b}\right\rbrack }\right) \] and so \[ \left| \left| {{\i... | Yes |
Lemma 50.0.6 Let \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{C} \) be in \( {C}^{1}\left( \left\lbrack {a, b}\right\rbrack \right) \) . Then \( V\left( {\gamma ,\left\lbrack {a, b}\right\rbrack }\right) < \infty \) so \( \gamma \) is of bounded variation. | Proof: This follows from the following\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{n}\left| {\gamma \left( {t}_{j}\right) - \gamma \left( {t}_{j - 1}\right) }\right| = \mathop{\sum }\limits_{{j = 1}}^{n}\left| {{\int }_{{t}_{j - 1}}^{{t}_{j}}{\gamma }^{\prime }\left( s\right) {ds}}\right| \]\n\n\[ \leq \mathop{\sum }\limits... | Yes |
Theorem 50.0.7 Let \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{C} \) be continuous and of bounded variation. Let \( \Omega \) be an open set containing \( {\gamma }^{ * } \) and let \( f : \Omega \times K \rightarrow X \) be continuous for \( K \) a compact set in \( \mathbb{C} \), and let \( \vare... | Proof: Extend \( \gamma \) to be defined on all \( \mathbb{R} \) according to \( \gamma \left( t\right) = \gamma \left( a\right) \) if \( t < a \) and \( \gamma \left( t\right) = \gamma \left( b\right) \) if \( t > b \) . Now define \[ {\gamma }_{h}\left( t\right) \equiv \frac{1}{2h}{\int }_{-{2h} + t + \frac{2h}{\left... | Yes |
Lemma 50.0.8 \( V\left( {{\gamma }_{h},\left\lbrack {a, b}\right\rbrack }\right) \leq V\left( {\gamma ,\left\lbrack {a, b}\right\rbrack }\right) \) . | Proof: Let \( a = {t}_{0} < {t}_{1} < \cdots < {t}_{n} = b \) . Then using the definition of \( {\gamma }_{h} \) and changing the variables to make all integrals over \( \left\lbrack {0,{2h}}\right\rbrack \) ,\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{n}\left| {{\gamma }_{h}\left( {t}_{j}\right) - {\gamma }_{h}\left( {t}_... | Yes |
Lemma 50.0.11 In the above definition, there exists a continuous bounded variation function, \( \gamma \) defined on some closed interval, \( \left\lbrack {c, d}\right\rbrack \), such that \( \gamma \left( \left\lbrack {c, d}\right\rbrack \right) = \) \( { \cup }_{k = 1}^{m}{\gamma }_{k}\left( \left\lbrack {{a}_{k},{b}... | \[ {\int }_{\gamma }f\left( z\right) {dz} = \mathop{\sum }\limits_{{k = 1}}^{m}{\int }_{{\gamma }_{k}}f\left( z\right) {dz}. \] | Yes |
Theorem 50.0.13 Let \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{C} \) be continuous and of bounded variation. Also suppose \( {F}^{\prime }\left( z\right) = f\left( z\right) \) for all \( z \in \Omega \), an open set containing \( {\gamma }^{ * } \) and \( f \) is continuous on \( \Omega \) . Then ... | Proof: By Theorem 50.0.12 there exists \( \eta \in {C}^{1}\left( \left\lbrack {a, b}\right\rbrack \right) \) such that \( \gamma \left( a\right) = \eta \left( a\right) \) , and \( \gamma \left( b\right) = \eta \left( b\right) \) such that \[ \begin{Vmatrix}{{\int }_{\gamma }f\left( z\right) {dz} - {\int }_{\eta }f\left... | Yes |
Corollary 50.0.14 If \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{C} \) is continuous, has bounded variation, is a closed curve, \( \gamma \left( a\right) = \gamma \left( b\right) \), and \( {\gamma }^{ * } \subseteq \Omega \) where \( \Omega \) is an open set on which \( {F}^{\prime }\left( z\right... | \[ {\int }_{\gamma }f\left( z\right) {dz} = 0 \] | Yes |
Theorem 51.1.3 Consider \( \mathop{\sum }\limits_{{k = 1}}^{\infty }{a}_{k} \) and let \( \rho \equiv \lim \mathop{\sup }\limits_{{k \rightarrow \infty }}{\begin{Vmatrix}{a}_{k}\end{Vmatrix}}^{1/k} \) . Then if \( \rho < \) 1, the series converges absolutely and if \( \rho > 1 \) the series diverges spectacularly in th... | Proof: Suppose \( \rho < 1 \) . Then there exists \( r \in \left( {\rho ,1}\right) \) . Therefore, \( \begin{Vmatrix}{a}_{k}\end{Vmatrix} \leq {r}^{k} \) for all \( k \) large enough and so by a comparison test, \( \mathop{\sum }\limits_{k}\begin{Vmatrix}{a}_{k}\end{Vmatrix} \) converges because the partial sums are bo... | Yes |
Lemma 51.1.6 Let \( \gamma \) denote the closed curve which is a circle of radius \( r \) centered at \( {z}_{0} \) . Then a parameterization this curve is \( \gamma \left( t\right) = {z}_{0} + r{e}^{it} \) where \( t \in \left\lbrack {0,{2\pi }}\right\rbrack \) . | Proof: \( {\left| \gamma \left( t\right) - {z}_{0}\right| }^{2} = \left| {r{e}^{it}r{e}^{-{it}}}\right| = {r}^{2} \) . Also, you can see from the definition of the sine and cosine that the point described in this way moves counter clockwise over this circle. | Yes |
Lemma 51.3.1 Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{C} \) . Then \( {f}^{\prime }\left( t\right) \) exists if and only if \( \operatorname{Re}{f}^{\prime }\left( t\right) \) and \( \operatorname{Im}{f}^{\prime }\left( t\right) \) exist. Furthermore, \[ {f}^{\prime }\left( t\right) = \operatorna... | Proof: The if part of the equivalence is obvious. Now suppose \( {f}^{\prime }\left( t\right) \) exists. Let both \( t \) and \( t + h \) be contained in \( \left\lbrack {a, b}\right\rbrack \) \[ \left| {\frac{\operatorname{Re}f\left( {t + h}\right) - \operatorname{Re}f\left( t\right) }{h} - \operatorname{Re}\left( {{f... | Yes |
Lemma 51.3.2 If \( g : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{C} \) and \( g \) is continuous on \( \left\lbrack {a, b}\right\rbrack \) and differentiable on \( \left( {a, b}\right) \) with \( {g}^{\prime }\left( t\right) = 0 \), then \( g\left( t\right) \) is a constant. | Proof: From the above lemma, you can apply the mean value theorem to the real and imaginary parts of \( g \) . | No |
Lemma 51.3.4 If \( g : \left\lbrack {a, b}\right\rbrack \rightarrow X \) and \( g \) is continuous on \( \left\lbrack {a, b}\right\rbrack \) and differentiable on \( \left( {a, b}\right) \) with \( {g}^{\prime }\left( t\right) = 0 \), then \( g\left( t\right) \) is a constant. | Proof: Let \( \Lambda \in {X}^{\prime } \) . Then \( {\Lambda g} : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{C} \) . Therefore, from Lemma 51.3.2, for each \( \Lambda \in {X}^{\prime },{\Lambda g}\left( s\right) = {\Lambda g}\left( t\right) \) and since \( {X}^{\prime } \) separates the points, it follows \(... | Yes |
Lemma 51.3.5 Let \( \phi : \left\lbrack {a, b}\right\rbrack \times \left\lbrack {c, d}\right\rbrack \rightarrow \mathbb{R} \) be continuous and let\n\n\[ g\left( t\right) \equiv {\int }_{a}^{b}\phi \left( {s, t}\right) {ds}. \]\n\nThen \( g \) is continuous. If \( \frac{\partial \phi }{\partial t} \) exists and is cont... | Proof: The first claim follows from the uniform continuity of \( \phi \) on \( \left\lbrack {a, b}\right\rbrack \times \left\lbrack {c, d}\right\rbrack \) , which uniform continuity results from the set being compact. To establish 51.3.2, let \( t \) and \( t + h \) be contained in \( \left\lbrack {c, d}\right\rbrack \... | Yes |
Corollary 51.3.6 Let \( \phi : \left\lbrack {a, b}\right\rbrack \times \left\lbrack {c, d}\right\rbrack \rightarrow \mathbb{C} \) be continuous and let\n\n\[ g\left( t\right) \equiv {\int }_{a}^{b}\phi \left( {s, t}\right) {ds}. \]\n\nThen \( g \) is continuous. If \( \frac{\partial \phi }{\partial t} \) exists and is ... | Proof: Apply Lemma 51.3.5 to the real and imaginary parts of \( \phi \) . | No |
Corollary 51.3.8 Let \( \phi : \left\lbrack {a, b}\right\rbrack \times \left\lbrack {c, d}\right\rbrack \rightarrow X \) be continuous and let\n\n\[ g\left( t\right) \equiv {\int }_{a}^{b}\phi \left( {s, t}\right) {ds}. \]\n\nThen \( g \) is continuous. If \( \frac{\partial \phi }{\partial t} \) exists and is continuou... | Proof: Let \( \Lambda \in {X}^{\prime } \) . Then \( {\Lambda \phi } : \left\lbrack {a, b}\right\rbrack \times \left\lbrack {c, d}\right\rbrack \rightarrow \mathbb{C} \) is continuous and \( \frac{\partial {\Lambda \phi }}{\partial t} \) exists and is continuous on \( \left\lbrack {a, b}\right\rbrack \times \left\lbrac... | Yes |
Theorem 51.3.9 Let \( \phi : \left\lbrack {a, b}\right\rbrack \times \left\lbrack {c, d}\right\rbrack \rightarrow X \) be continuous and suppose \( {\phi }_{t} \) is continuous. Then \[ {\left( {\int }_{a}^{b}\phi \left( s, t\right) ds\right) }_{, t} = {\int }_{a}^{b}\frac{\partial \phi }{\partial t}\left( {s, t}\right... | Proof: Consider the following set \( P \) which is where the ordered pair \( \left( {t, h}\right) \) will be.  This is so that both \( t \) and \( t + h \) are in \( \left\lbrack {a, b}\right\rbrack \) . Then for s... | Yes |
Theorem 51.3.10 Let \( f : \Omega \rightarrow X \) be analytic on the open set, \( \Omega \) and let\n\n\[ \overline{B\left( {{z}_{0}, r}\right) } \subseteq \Omega \text{.} \]\n\nLet \( \gamma \left( t\right) \equiv {z}_{0} + r{e}^{it} \) for \( t \in \left\lbrack {0,{2\pi }}\right\rbrack \) . Then if \( z \in B\left( ... | Proof: Consider for \( \alpha \in \left\lbrack {0,1}\right\rbrack \), \n\n\[ g\left( \alpha \right) \equiv {\int }_{0}^{2\pi }\frac{f\left( {z + \alpha \left( {{z}_{0} + r{e}^{it} - z}\right) }\right) }{r{e}^{it} + {z}_{0} - z}{ri}{e}^{it}{dt}. \]\n\nIf \( \alpha \) equals one, this reduces to the integral in 51.3.9. T... | Yes |
Theorem 51.3.11 Let \( f : \Omega \rightarrow X \) be analytic where \( \Omega \) is an open set in \( \mathbb{C} \) . Then \( f \) has infinitely many derivatives on \( \Omega \) . Furthermore, for all \( z \in B\left( {{z}_{0}, r}\right) \) , \[ {f}^{\left( n\right) }\left( z\right) = \frac{n!}{2\pi i}{\int }_{\gamma... | Proof: Let \( z \in B\left( {{z}_{0}, r}\right) \subseteq \Omega \) and let \( \overline{B\left( {{z}_{0}, r}\right) } \subseteq \Omega \) . Then, letting \( \gamma \left( t\right) \equiv \) \( {z}_{0} + r{e}^{it}, t \in \left\lbrack {0,{2\pi }}\right\rbrack \), and \( h \) small enough, \[ f\left( z\right) = \frac{1}{... | Yes |
Lemma 51.3.13 Let \( \gamma \left( t\right) = {z}_{0} + r{e}^{it} \), for \( t \in \left\lbrack {0,{2\pi }}\right\rbrack \), suppose \( {f}_{n} \rightarrow f \) uniformly on \( \overline{B\left( {{z}_{0}, r}\right) } \), and suppose\n\n\[ \n{f}_{n}\left( z\right) = \frac{1}{2\pi i}{\int }_{\gamma }\frac{{f}_{n}\left( w... | Proof: From 51.3.11 and the uniform convergence of \( {f}_{n} \) to \( f \) on \( \gamma \left( \left\lbrack {0,{2\pi }}\right\rbrack \right) \), the integrals in 51.3.11 converge to\n\n\[ \n\frac{1}{2\pi i}{\int }_{\gamma }\frac{f\left( w\right) }{w - z}{dw} \n\]\n\nTherefore, the formula 51.3.12 follows. | Yes |
Proposition 51.3.14 Let \( \\left\\{ {a}_{n}\\right\\} \) denote a sequence in \( X \) . Then there exists \( R \\in \) \( \\left\\lbrack {0,\\infty }\\right\\rbrack \) such that\n\n\[ \n\\mathop{\\sum }\\limits_{{k = 0}}^{\\infty }{a}_{k}{\\left( z - {z}_{0}\\right) }^{k} \n\]\n\nconverges absolutely if \( \\left| {z ... | Proof: The assertions about absolute convergence are routine from the root test if\n\n\[ \nR \\equiv {\\left( \\lim \\mathop{\\sup }\\limits_{{n \\rightarrow \\infty }}{\\left| {a}_{n}\\right| }^{1/n}\\right) }^{-1} \n\]\n\nwith \( R = \\infty \) if the quantity in parenthesis equals zero. The root test can be used to ... | Yes |
Theorem 51.3.15 If \( f : \Omega \rightarrow X \) is analytic and if \( B\left( {{z}_{0}, r}\right) \subseteq \Omega \), then\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{\left( z - {z}_{0}\right) }^{n} \]\n\nfor all \( \left| {z - {z}_{0}}\right| < r \) . Furthermore,\n\n\[ {a}_{n} = \frac... | Proof: Consider \( \left| {z - {z}_{0}}\right| < r \) and let \( \gamma \left( t\right) = {z}_{0} + r{e}^{it}, t \in \left\lbrack {0,{2\pi }}\right\rbrack \) . Then for \( w \in \gamma \left( \left\lbrack {0,{2\pi }}\right\rbrack \right) ,\n\n\[ \left| \frac{z - {z}_{0}}{w - {z}_{0}}\right| < 1 \]\nand so, by the Cauch... | Yes |
Theorem 51.5.3 Let \( \Omega \) be a connected open set (region) and let \( f : \Omega \rightarrow X \) be analytic. Then the following are equivalent.\n\n1. \( f\left( z\right) = 0 \) for all \( z \in \Omega \)\n\n2. There exists \( {z}_{0} \in \Omega \) such that \( {f}^{\left( n\right) }\left( {z}_{0}\right) = 0 \) ... | Proof: It is clear the first condition implies the second two. Suppose the third holds. Then for \( z \) near \( {z}_{0} \)\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{n = k}}^{\infty }\frac{{f}^{\left( n\right) }\left( {z}_{0}\right) }{n!}{\left( z - {z}_{0}\right) }^{n} \]\n\nwhere \( k \geq 1 \) since \( {z}_{0... | Yes |
Theorem 51.5.4 (Euler’s Formula) Let \( z = x + {iy} \) . Then\n\n\[ {e}^{z} = \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{{z}^{k}}{k!} \] | Proof: It was already observed that \( {e}^{z} \) given by 51.5.19 is analytic. So is \( \exp \left( z\right) \equiv \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{{z}^{k}}{k!} \) . In fact the power series converges for all \( z \in \mathbb{C} \) . Furthermore the two functions, \( {e}^{z} \) and \( \exp \left( z\righ... | Yes |
Theorem 51.6.2 (Liouville's theorem) If \( f \) is a bounded entire function having values in \( X \), then \( f \) is a constant. | Proof: Since \( f \) is entire, pick any \( z \in \mathbb{C} \) and write\n\n\[ \n{f}^{\prime }\left( z\right) = \frac{1}{2\pi i}{\int }_{{\gamma }_{R}}\frac{f\left( w\right) }{{\left( w - z\right) }^{2}}{dw} \n\]\n\nwhere \( {\gamma }_{R}\left( t\right) = z + R{e}^{it} \) for \( t \in \left\lbrack {0,{2\pi }}\right\rb... | Yes |
Theorem 51.6.3 (Fundamental theorem of Algebra) Let\n\n\\[ \np\\left( z\\right) = {z}^{n} + {a}_{n - 1}{z}^{n - 1} + \\cdots + {a}_{1}z + {a}_{0} \n\\]\n\nbe a polynomial where \\( n \\geq 1 \\) and each coefficient is a complex number. Then there exists \\( {z}_{0} \\in \\mathbb{C} \\) such that \\( p\\left( {z}_{0}\\... | Proof: Suppose not. Then \\( p{\\left( z\\right) }^{-1} \\) is an entire function. Also\n\n\\[ \n\\left| {p\\left( z\\right) }\\right| \\geq {\\left| z\\right| }^{n} - \\left( {\\left| {a}_{n - 1}\\right| {\\left| z\\right| }^{n - 1} + \\cdots + \\left| {a}_{1}\\right| \\left| z\\right| + \\left| {a}_{0}\\right| }\\rig... | Yes |
Theorem 51.7.1 (Cauchy Goursat) Let \( f : \Omega \rightarrow X \) have the property that \( {f}^{\prime }\left( z\right) \) exists for all \( z \in \Omega \) and let \( T \) be a triangle contained in \( \Omega \) . Then\n\n\[{\int }_{\partial T}f\left( w\right) {dw} = 0\] | Proof: Suppose not. Then\n\n\[ \left| \left| {{\int }_{\partial T}f\left( w\right) {dw}}\right| \right| = \alpha \neq 0 \]\n\nFrom 51.7.20 it follows\n\n\[ \alpha \leq \mathop{\sum }\limits_{{k = 1}}^{4}\begin{Vmatrix}{{\int }_{\partial {T}_{k}^{1}}f\left( w\right) {dw}}\end{Vmatrix} \]\n\nand so for at least one of th... | Yes |
Theorem 51.7.2 (Morera \( {}^{1} \) ) Let \( \Omega \) be an open set and let \( {f}^{\prime }\left( z\right) \) exist for all \( z \in \Omega \) . Let \( D \equiv \overline{B\left( {{z}_{0}, r}\right) } \subseteq \Omega \) . Then there exists \( \varepsilon > 0 \) such that \( f \) has a primitive on \( B\left( {{z}_{... | Proof: Choose \( \varepsilon > 0 \) small enough that \( B\left( {{z}_{0}, r + \varepsilon }\right) \subseteq \Omega \) . Then for \( w \in \) \( B\left( {{z}_{0}, r + \varepsilon }\right) \), define\n\n\[ F\left( w\right) \equiv {\int }_{\gamma \left( {{z}_{0}, w}\right) }f\left( u\right) {du}. \]\n\nThen by the Cauch... | Yes |
Corollary 51.7.3 Let \( \Omega \) be an open set and suppose that whenever\n\n\[ \n\gamma \left( {{z}_{1},{z}_{2},{z}_{3},{z}_{1}}\right)\n\]\n\nis a closed curve bounding a triangle \( T \), which is contained in \( \Omega \), and \( f \) is a continuous function defined on \( \Omega \), it follows that\n\n\[ \n{\int ... | Proof: As in the proof of Morera’s theorem, let \( \overline{B\left( {{z}_{0}, r}\right) } \subseteq \Omega \) and use the given condition to construct a primitive, \( F \) for \( f \) on \( B\left( {{z}_{0}, r}\right) \) . Then \( F \) is analytic and so by Theorem 51.3.11, it follows that \( F \) and hence \( f \) ha... | Yes |
Theorem 51.7.4 Let \( \Omega \) be an open set in \( \mathbb{C} \) and suppose \( f : \Omega \rightarrow X \) has the property that \( {f}^{\prime }\left( z\right) \) exists for each \( z \in \Omega \) . Then \( f \) is analytic on \( \Omega \) . | Proof: Let \( {z}_{0} \in \Omega \) and let \( B\left( {{z}_{0}, r}\right) \subseteq \Omega \) . By Morera’s theorem \( f \) has a primitive, \( F \) on \( B\left( {{z}_{0}, r}\right) \) . It follows that \( F \) is analytic because it has a derivative, \( f \) , and this derivative is continuous. Therefore, by Theorem... | Yes |
Corollary 51.7.5 Let \( \Omega \) be a convex open set and suppose that \( {f}^{\prime }\left( z\right) \) exists for all \( z \in \Omega \) . Then \( f \) has a primitive on \( \Omega \) . | Note that this implies that if \( \Omega \) is a convex open set on which \( {f}^{\prime }\left( z\right) \) exists and if \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow \Omega \) is a closed, continuous curve having bounded variation, then letting \( F \) be a primitive of \( f \) Theorem 50.0.13 implies\n\n... | No |
Theorem 51.7.8 Let \( f : {B}^{\prime }\left( {a, r}\right) \rightarrow X \) be analytic. Thus \( f \) has an isolated singularity at a. Suppose also that \[ \mathop{\lim }\limits_{{z \rightarrow a}}f\left( z\right) \left( {z - a}\right) = 0. \] Then there exists a unique analytic function, \( g : B\left( {a, r}\right)... | Proof: Let \( h\left( z\right) \equiv {\left( z - a\right) }^{2}f\left( z\right), h\left( a\right) \equiv 0 \) . Then \( h \) is analytic on \( B\left( {a, r}\right) \) because it is easy to see that \( {h}^{\prime }\left( a\right) = 0 \) . It follows \( h \) is given by a power series, \[ h\left( z\right) = \mathop{\s... | Yes |
Theorem 51.7.9 (Casorati Weierstrass) Let a be an isolated singularity and suppose for some \( r > 0, f\left( {{B}^{\prime }\left( {a, r}\right) }\right) \) is not dense in \( \mathbb{C} \) . Then either \( a \) is a removable singularity or there exist finitely many \( {b}_{1},\cdots ,{b}_{M} \) for some finite number... | Proof: Suppose \( B\left( {{z}_{0},\delta }\right) \) has no points of \( f\left( {{B}^{\prime }\left( {a, r}\right) }\right) \) . Such a ball must exist if \( f\left( {{B}^{\prime }\left( {a, r}\right) }\right) \) is not dense. Then for \( z \in {B}^{\prime }\left( {a, r}\right) ,\left| {f\left( z\right) - {z}_{0}}\ri... | Yes |
Theorem 51.7.11 Suppose \( f : \Omega \rightarrow \mathbb{C} \) has an isolated singularity at \( a \in \Omega \) . Then a is a pole if and only if\n\n\[ \mathop{\lim }\limits_{{z \rightarrow a}}d\left( {f\left( z\right) ,\infty }\right) = 0 \]\n\nin \( \widehat{\mathbb{C}} \) . | Proof: Suppose first \( f \) has a pole at \( a \) . Then by definition, \( f\left( z\right) = g\left( z\right) + \) \( \mathop{\sum }\limits_{{k = 1}}^{M}\frac{{b}_{k}}{{\left( z - a\right) }^{k}} \) for \( z \) near \( a \) where \( g \) is analytic. Then\n\n\[ \left| {f\left( z\right) }\right| \geq \frac{\left| {b}_... | Yes |
Theorem 51.7.13 Let \( \Omega \) be an open subset of \( \mathbb{C} \) and let \( f : \Omega \rightarrow \widehat{\mathbb{C}} \) be meromorphic. Then \( f \) is continuous with respect to the metric, \( d \) on \( \widehat{\mathbb{C}} \) . | Proof: Let \( {z}_{n} \rightarrow z \) where \( z \in \Omega \) . Then if \( z \) is a pole, it follows from Theorem 51.7.11 that\n\n\[ d\left( {f\left( {z}_{n}\right) ,\infty }\right) \equiv d\left( {f\left( {z}_{n}\right), f\left( z\right) }\right) \rightarrow 0. \]\n\nIf \( z \) is not a pole, then \( f\left( {z}_{n... | Yes |
Theorem 51.7.15 Let \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{C} \) be continuous and have bounded variation with \( \gamma \left( a\right) = \gamma \left( b\right) \) . Also suppose that \( z \notin {\gamma }^{ * } \) . Define\n\n\[ n\left( {\gamma, z}\right) \equiv \frac{1}{2\pi i}{\int }_{\gam... | Proof: First consider the assertion about continuity.\n\n\[ \left| {n\left( {\gamma, z}\right) - n\left( {\gamma ,{z}_{1}}\right) }\right| \leq C\left| {{\int }_{\gamma }\left( {\frac{1}{w - z} - \frac{1}{w - {z}_{1}}}\right) {dw}}\right| \]\n\n\[ \leq \widetilde{C}\text{(Length of}\gamma \text{)}\left| {{z}_{1} - z}\r... | Yes |
Corollary 51.7.16 Suppose \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{C} \) is a continuous bounded variation curve and \( n\left( {\gamma, z}\right) \) is an integer where \( z \notin {\gamma }^{ * } \) . Then \( \gamma \left( a\right) = \gamma \left( b\right) \) . Also \( z \rightarrow n\left( {\... | Proof: Letting \( \eta \) be a \( {C}^{1} \) curve for which \( \eta \left( a\right) = \gamma \left( a\right) \) and \( \eta \left( b\right) = \gamma \left( b\right) \) and which is close enough to \( \gamma \) that \( n\left( {\eta, z}\right) = n\left( {\gamma, z}\right) \), the argument is similar to the above. Let \... | Yes |
Proposition 51.7.17 Under the above conditions, \[ n\left( {\delta, z}\right) = n\left( {\gamma, z}\right) \] and \( n\left( {\delta, z}\right) = 1 \) . | Proof: By changing the parameter, assume that \( \left\lbrack {a, b}\right\rbrack = \left\lbrack {0,{2\pi }}\right\rbrack \) . From Theorem 51.7.15 it suffices to assume also that \( \gamma \) is \( {C}^{1} \) . Define \( {h}_{\lambda }\left( t\right) \equiv \gamma \left( t\right) + \lambda \left( {z + r{e}^{it} - \gam... | Yes |
Corollary 51.7.20 Let \( \Omega \) be an open set and let \( {\gamma }_{k} : \left\lbrack {{a}_{k},{b}_{k}}\right\rbrack \rightarrow \Omega, k = 1,\cdots, m \) , be closed, continuous and of bounded variation. Suppose also that\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{m}n\left( {{\gamma }_{k}, z}\right) = 0 \]\n\nfor all... | Proof: This follows from Theorem 51.7.19 as follows. Let\n\n\[ g\left( w\right) = f\left( w\right) \left( {w - z}\right) \]\n\nwhere \( z \in \Omega \smallsetminus { \cup }_{k = 1}^{m}{\gamma }_{k}\left( \left\lbrack {{a}_{k},{b}_{k}}\right\rbrack \right) \) . Then by this theorem,\n\n\[ 0 = 0\mathop{\sum }\limits_{{k ... | Yes |
Corollary 51.7.22 Let \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow \Omega \) be a continuous closed curve of bounded variation where \( \Omega \) is a simply connected region in \( \mathbb{C} \) and let \( f : \Omega \rightarrow X \) be analytic.\n\nThen\n\[{\int }_{\gamma }f\left( w\right) {dw} = 0\] | Proof: Let \( D \) denote the unbounded component of \( \widehat{\mathbb{C}} \smallsetminus {\gamma }^{ * } \) . Thus \( \infty \in \widehat{\mathbb{C}} \smallsetminus {\gamma }^{ * } \) . Then the connected set, \( \widehat{\mathbb{C}} \smallsetminus \Omega \) is contained in \( D \) since every point of \( \widehat{\... | Yes |
Corollary 51.7.23 Suppose \( \Omega \) is a simply connected open set and \( f : \Omega \rightarrow X \) is analytic. Then \( f \) has a primitive, \( F \), on \( \Omega \) . Recall this means there exists \( F \) such that \( {F}^{\prime }\left( z\right) = f\left( z\right) \) for all \( z \in \Omega \) . | Proof: Pick a point, \( {z}_{0} \in \Omega \) and let \( V \) denote those points, \( z \) of \( \Omega \) for which there exists a curve, \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow \Omega \) such that \( \gamma \) is continuous, of bounded variation, \( \gamma \left( a\right) = {z}_{0} \), and \( \gamma ... | Yes |
Corollary 52.1.2 Suppose in the situation of Theorem 52.1.1 \( m > 1 \) for the local representation of \( f \) given in this theorem. Then there exists \( \delta > 0 \) such that if \( w \in B\left( {f\left( {z}_{0}\right) ,\delta }\right) = f\left( V\right) \) for \( V \) an open set containing \( {z}_{0} \), then \(... | Proof: Let \( w \in B\left( {f\left( {z}_{0}\right) ,\delta }\right) \) . Then \( w = f\left( \widehat{z}\right) \) where \( \widehat{z} \in V \) . Thus \( f\left( \widehat{z}\right) = \) \( f\left( {z}_{0}\right) + \phi {\left( \widehat{z}\right) }^{m} \) . Consider the \( m \) distinct numbers, \( {\left\{ {e}^{\frac... | Yes |
Theorem 52.2.1 Let \( \Omega \) be a simply connected region and suppose \( f : \Omega \rightarrow \mathbb{C} \) is analytic and nonzero on \( \Omega \) . Then there exists an analytic function, \( g \) such that \( {e}^{g\left( z\right) } = f\left( z\right) \) for all \( z \in \Omega \) . | Proof: The function, \( {f}^{\prime }/f \) is analytic on \( \Omega \) and so by Corollary 51.7.23 there is a primitive for \( {f}^{\prime }/f \), denoted as \( {g}_{1} \) . Then\n\n\[ \n{\left( {e}^{-{g}_{1}}f\right) }^{\prime } = - \frac{{f}^{\prime }}{f}{e}^{-{g}_{1}}f + {e}^{-{g}_{1}}{f}^{\prime } = 0 \n\] \n\nand ... | Yes |
Theorem 52.2.3 Let \( \rho \) be a ray starting at 0 . Then there exists an analytic function, \( L\left( z\right) \) defined on \( \mathbb{C} \smallsetminus \rho \) such that\n\n\[ \n{e}^{L\left( z\right) } = z.\n\]\n\nThis function, \( L \) is called a branch of the logarithm. This branch of the logarithm satisfies t... | Proof: \( \mathbb{C} \smallsetminus \rho \) is a simply connected region because its complement with respect to \( \widehat{\mathbb{C}} \) is connected. Furthermore, the function, \( f\left( z\right) = z \) is not equal to zero on \( \mathbb{C} \smallsetminus \rho \) . Therefore, by Theorem 52.2.1 there exists an analy... | Yes |
Theorem 52.3.1 (maximum modulus theorem) Let \( \Omega \) be a bounded region and let \( f : \Omega \rightarrow \mathbb{C} \) be analytic and \( f : \bar{\Omega } \rightarrow \mathbb{C} \) continuous. Then if \( z \in \Omega \) , \[ \left| {f\left( z\right) }\right| \leq \max \{ \left| {f\left( w\right) }\right| : w \i... | Proof: Suppose \( f \) is not a constant. Then \( f\left( \Omega \right) \) is a region and so if \( z \in \Omega \) , there exists \( r > 0 \) such that \( B\left( {f\left( z\right), r}\right) \subseteq f\left( \Omega \right) \) . It follows there exists \( {z}_{1} \in \Omega \) with \( \left| {f\left( {z}_{1}\right) ... | Yes |
Theorem 52.3.2 Let \( f : \Omega \rightarrow \mathbb{C} \) be analytic on a region, \( \Omega \) and suppose \( \overline{B\left( {a, r}\right) } \subseteq \) \( \Omega \) . Then\n\n\[ \left| {f\left( a\right) }\right| \leq \max \left\{ {\left| {f\left( {a + r{e}^{i\theta }}\right) }\right| : \theta \in \left\lbrack {0... | Proof: The claimed inequality holds by Theorem 52.3.1. Suppose equality in the above is achieved for some \( \overline{B\left( {a, r}\right) } \subseteq \Omega \) . Then by Theorem 52.3.1 \( f \) is equal to a constant, \( w \) on \( B\left( {a, r}\right) \) . Therefore, the function, \( f\left( \cdot \right) - w \) ha... | Yes |
Theorem 52.3.5 Let \( \Omega \) be an open set in \( \mathbb{C} \) and let \( f : \Omega \rightarrow \mathbb{C} \) be analytic. Suppose also that for every \( a \in {\partial }_{\infty }\Omega \) , \[ \lim \mathop{\sup }\limits_{{z \rightarrow a}}\left| {f\left( z\right) }\right| \leq M < \infty . \] Then in fact \( \l... | Proof: Let \( \delta > 0 \) and let \( H \equiv \{ z \in \Omega : \left| {f\left( z\right) }\right| > M + \delta \} \) . Suppose \( H \neq \varnothing \) . Then \( H \) is an open subset of \( \Omega \) . I claim that \( H \) is actually bounded. If \( \Omega \) is bounded, there is nothing to show so assume \( \Omega ... | Yes |
Theorem 52.4.1 Let \( \Omega \) be a simply connected region in \( \mathbb{C} \) and suppose \( f \) is analytic on \( \Omega \) . Also suppose there exists a function, \( \phi \) which is nonzero and uniformly bounded on \( \Omega \) . Let \( M \) be a positive number. Now suppose \( {\partial }_{\infty }\Omega = A \c... | Proof: By Theorem 52.2.1 there exists \( \log \left( {\phi \left( z\right) }\right) \) analytic on \( \Omega \) . Now define \( g\left( z\right) \equiv \exp \left( {\eta \log \left( {\phi \left( z\right) }\right) }\right) \) so that \( g\left( z\right) = \phi {\left( z\right) }^{\eta } \) . Now also\n\n\[ \left| {g\lef... | Yes |
Corollary 52.4.2 Let \( \Omega = \left\{ {z \in \mathbb{C} : \left| {\arg \left( z\right) }\right| < \frac{\pi }{2a}}\right\} \) where \( a \geq \frac{1}{2} \) and suppose \( f \) is analytic on \( \Omega \) and satisfies \( \mathop{\limsup }\limits_{{z \rightarrow a}}\left| {f\left( z\right) }\right| \leq M \) on \( \... | Proof: Let \( b < c < a \) and let \( \phi \left( z\right) \equiv \exp \left( {-\left( {z}^{c}\right) }\right) \) . Then as discussed above, \( \phi \left( z\right) \neq 0 \) on \( \Omega \) and \( \left| {\phi \left( z\right) }\right| \) is bounded on \( \Omega \) . Now\n\n\[ {\left| \phi \left( z\right) \right| }^{\e... | Yes |
Corollary 52.4.3 Let \( \Omega \) be the open set consisting of \( \{ z \in \mathbb{C} : a < \operatorname{Re}z < b\} \) and suppose \( f \) is analytic on \( \Omega \), continuous on \( \bar{\Omega } \), and bounded on \( \Omega \) . Suppose also that \( f\left( z\right) \leq 1 \) on the two lines \( \operatorname{Re}... | Proof: This time let \( \phi \left( z\right) = \frac{1}{1 + z - a} \) . Thus \( \left| {\phi \left( z\right) }\right| \leq 1 \) because \( \operatorname{Re}\left( {z - a}\right) > 0 \) and \( \phi \left( z\right) \neq 0 \) for all \( z \in \Omega \) . Also, \( \lim \mathop{\sup }\limits_{{z \rightarrow \infty }}{\left|... | Yes |
Corollary 52.4.4 Let \( \Omega \) be the open set consisting of \( \{ z \in \mathbb{C} : a < \operatorname{Re}z < b\} \) and suppose \( f \) is analytic on \( \Omega \), continuous on \( \bar{\Omega } \), and bounded on \( \Omega \) . Define\n\n\[ M\left( x\right) \equiv \sup \{ \left| {f\left( z\right) }\right| : \ope... | Proof: Let \( \varepsilon > 0 \) and define\n\n\[ g\left( z\right) \equiv {\left( M\left( a\right) + \varepsilon \right) }^{\frac{b - z}{b - a}}{\left( M\left( b\right) + \varepsilon \right) }^{\frac{z - a}{b - a}} \]\n\nwhere for \( M > 0 \) and \( z \in \mathbb{C},{M}^{z} \equiv \exp \left( {z\ln \left( M\right) }\ri... | Yes |
Corollary 52.4.5 Let \( \Omega = \\left\\{ {z \\in \\mathbb{C} : \\left| {\\operatorname{Im}\\left( z\\right) }\\right| < \\frac{\\pi }{2}}\\right\\} \) . Suppose \( f \) is analytic on \( \\Omega \) , continuous on \( \\bar{\\Omega } \), and there exist constants, \( \\alpha < 1 \) and \( A < \\infty \) such that\n\n\... | Proof: This time let \( \\phi \\left( z\\right) = {\\left\\lbrack \\exp \\left( A\\exp \\left( \\beta z\\right) \\right) \\exp \\left( A\\exp \\left( -\\beta z\\right) \\right) \\right\\rbrack }^{-1} \) where \( \\alpha < \\beta < 1 \) . Then \( \\phi \\left( z\\right) \\neq 0 \) on \( \\Omega \) and for \( \\eta > 0 \... | Yes |
Lemma 52.4.6 Suppose \( F : B\left( {0,1}\right) \rightarrow B\left( {0,1}\right), F \) is analytic, and \( F\left( 0\right) = 0 \) . Then for all \( z \in B\left( {0,1}\right) \) ,\n\n\[ \left| {F\left( z\right) }\right| \leq \left| z\right| \]\n\n\( \left( {52.4.4}\right) \)\n\nand\n\n\[ \left| {{F}^{\prime }\left( 0... | Proof: First note that by assumption, \( F\left( z\right) /z \) has a removable singularity at 0 if its value at 0 is defined to be \( {F}^{\prime }\left( 0\right) \) . By the maximum modulus theorem, if \( \left| z\right| < r < 1 \)\n\n\[ \left| \frac{F\left( z\right) }{z}\right| \leq \mathop{\max }\limits_{{t \in \le... | Yes |
Lemma 52.4.7 Let \( \alpha \in B\left( {0,1}\right) \) and define\n\n\[{\phi }_{\alpha }\left( z\right) \equiv \frac{z - \alpha }{1 - \bar{\alpha }z}.\n\]\n\nThen \( {\phi }_{\alpha } : B\left( {0,1}\right) \rightarrow B\left( {0,1}\right) ,{\phi }_{\alpha } : \partial B\left( {0,1}\right) \rightarrow \partial B\left( ... | Proof: First of all, for \( \left| z\right| < 1/\left| \alpha \right| \) ,\n\n\[{\phi }_{\alpha } \circ {\phi }_{-\alpha }\left( z\right) \equiv \frac{\left( \frac{z + \alpha }{1 + \bar{\alpha }z}\right) - \alpha }{1 - \bar{\alpha }\left( \frac{z + \alpha }{1 + \bar{\alpha }z}\right) } = z\n\]\n\nafter a few computatio... | Yes |
Theorem 52.4.8 Suppose \( f \) is an analytic function defined on \( B\left( {0,1}\right) \) and \( f \) maps \( B\left( {0,1}\right) \) one to one and onto \( B\left( {0,1}\right) \) . Then there exists \( \theta \) such that\n\n\[ f\left( z\right) = {e}^{i\theta }{\phi }_{\alpha }\left( z\right) \]\n\nfor some \( \al... | Proof: Let \( f\left( \alpha \right) = 0 \) . Then \( h\left( z\right) \equiv f \circ {\phi }_{-\alpha }\left( z\right) \) maps \( B\left( {0,1}\right) \) one to one and onto \( B\left( {0,1}\right) \) and has the property that \( h\left( 0\right) = 0 \) . Therefore, by the Schwarz lemma,\n\n\[ \left| {h\left( z\right)... | Yes |
Theorem 52.6.1 Let \( \Omega \) be an open set in \( \mathbb{C} \) and let \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow \Omega \) be closed, continuous, bounded variation, and \( n\left( {\gamma, z}\right) = 0 \) for all \( z \notin \Omega \) . Suppose also that \( f \) is analytic on \( \Omega \) having ze... | Proof: Let \( f\left( z\right) = \mathop{\prod }\limits_{{j = 1}}^{m}\left( {z - {a}_{j}}\right) g\left( z\right) \) where \( g\left( z\right) \neq 0 \) on \( \Omega \) . Note that some of the \( {a}_{j} \) could be repeated. Hence\n\n\[ \frac{{f}^{\prime }\left( z\right) }{f\left( z\right) } = \mathop{\sum }\limits_{{... | Yes |
Theorem 52.6.4 (open mapping theorem) Let \( \Omega \) be a region and \( f : \Omega \rightarrow \mathbb{C} \) be analytic. Then \( f\left( \Omega \right) \) is either a point or a region. If \( f \) is one to one, then \( {f}^{-1} : f\left( \Omega \right) \rightarrow \Omega \) is analytic. | Proof: If \( f \) is not constant, then for every \( \alpha \in f\left( \Omega \right) \), it follows from Theorem 51.5.3 that \( f\left( \cdot \right) - \alpha \) has a zero of order \( m < \infty \) and so from Theorem 52.6.3, for each \( a \in \Omega \) there exist \( \varepsilon ,\delta > 0 \) such that \( f\left( ... | Yes |
Theorem 52.7.1 Let \( A \) be an \( n \times n \) matrix. Consider the \( n \) Gerschgorin discs defined as\n\n\[ \n{D}_{i} \equiv \left\{ {\lambda \in \mathbb{C} : \left| {\lambda - {a}_{ii}}\right| \leq \mathop{\sum }\limits_{{j \neq i}}\left| {a}_{ij}\right| }\right\} .\n\]\n\nThen every eigenvalue is contained in s... | Proof: Suppose \( A\mathbf{x} = \lambda \mathbf{x} \) where \( \mathbf{x} \neq \mathbf{0} \) . Then for \( A = \left( {a}_{ij}\right) \)\n\n\[ \n\mathop{\sum }\limits_{{j \neq i}}{a}_{ij}{x}_{j} = \left( {\lambda - {a}_{ii}}\right) {x}_{i}\n\]\n\nTherefore, if we pick \( k \) such that \( \left| {x}_{k}\right| \geq \le... | Yes |
Theorem 52.7.4 Suppose \( A\left( t\right) \) is an \( n \times n \) matrix and that \( t \rightarrow A\left( t\right) \) is continuous for \( t \in \left\lbrack {0,1}\right\rbrack \) . Let \( \lambda \left( 0\right) \in \sigma \left( {A\left( 0\right) }\right) \) and define \( \sum \equiv { \cup }_{t \in \left\lbrack ... | Proof: Let \( S \equiv \left\{ {t \in \left\lbrack {0,1}\right\rbrack : {K}_{0} \cap \sigma \left( {A\left( s\right) }\right) \neq \varnothing }\right. \) for all \( \left. {s \in \left\lbrack {0, t}\right\rbrack }\right\} \) . Then \( 0 \in S \) . Let \( {t}_{0} = \sup \left( S\right) \) . Say \( \sigma \left( {A\left... | Yes |
Corollary 52.7.5 Suppose one of the Gerschgorin discs, \( {D}_{i} \) is disjoint from the union of the others. Then \( {D}_{i} \) contains an eigenvalue of \( A \) . Also, if there are \( n \) disjoint Gerschgorin discs, then each one contains an eigenvalue of \( A \) . | Proof: Denote by \( A\left( t\right) \) the matrix \( \left( {a}_{ij}^{t}\right) \) where if \( i \neq j,{a}_{ij}^{t} = t{a}_{ij} \) and \( {a}_{ii}^{t} = {a}_{ii} \) . Thus to get \( A\left( t\right) \) we multiply all non diagonal terms by \( t \) . Let \( t \in \left\lbrack {0,1}\right\rbrack \) . Then \( A\left( 0\... | Yes |
Corollary 52.7.7 Suppose one of the Gerschgorin discs, \( {D}_{i} \) is disjoint from the union of the others. Then \( {D}_{i} \) contains exactly one eigenvalue of \( A \) and this eigenvalue is a simple root to the characteristic polynomial of \( A \) . | Proof: In the proof of Corollary 52.7.5, first note that \( {a}_{ii} \) is a simple root of \( A\left( 0\right) \) since otherwise the \( {i}^{\text{th }} \) Gerschgorin disc would not be disjoint from the others. Also, \( K \), the connected component determined by \( {a}_{ii} \) must be contained in \( {D}_{i} \) bec... | Yes |
Example 52.7.8 Consider the matrix,\n\n\[ \left( \begin{array}{lll} 5 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 0 \end{array}\right) \]\n\nThe Gerschgorin discs are \( D\left( {5,1}\right), D\left( {1,2}\right) \), and \( D\left( {0,1}\right) \) . Then \( D\left( {5,1}\right) \) is disjoint from the other discs. Therefore, there... | The actual eigenvalues are not easy to find. They are the roots of the characteristic equation, \( {t}^{3} - 6{t}^{2} + {3t} + 5 = 0 \) . The numerical values of these are \( - {.66966},{1.4231} \) , and 5.24655 , verifying the predictions of Gerschgorin's theorem. | Yes |
Find \( \mathop{\lim }\limits_{{R \rightarrow \infty }}{\int }_{-R}^{R}\frac{\sin \left( x\right) }{x}{dx} \) | Things are easier if you write it as\n\n\[ \mathop{\lim }\limits_{{R \rightarrow \infty }}\frac{1}{i}\left( {{\int }_{-R}^{-{R}^{-1}}\frac{{e}^{ix}}{x}{dx} + {\int }_{{R}^{-1}}^{R}\frac{{e}^{ix}}{x}{dx}}\right) .\n\]\n\nThis gives the same answer because \( \cos \left( x\right) /x \) is odd. Consider the following cont... | Yes |
Find \( \mathop{\lim }\limits_{{R \rightarrow \infty }}{\int }_{-R}^{R}{e}^{ixt}\frac{\sin x}{x}{dx} \) . Note this is essentially finding the inverse Fourier transform of the function, \( \sin \left( x\right) /x \) . | This equals\n\n\[ \mathop{\lim }\limits_{{R \rightarrow \infty }}{\int }_{-R}^{R}\left( {\cos \left( {xt}\right) + i\sin \left( {xt}\right) }\right) \frac{\sin \left( x\right) }{x}{dx} \]\n\n\[ = \mathop{\lim }\limits_{{R \rightarrow \infty }}{\int }_{-R}^{R}\cos \left( {xt}\right) \frac{\sin \left( x\right) }{x}{dx} \... | Yes |
Theorem 53.1.1 (argument principle) Let \( f \) be meromorphic in \( \Omega \) . Also suppose \( {\gamma }^{ * } \) is a closed bounded variation curve containing none of the poles or zeros of \( f \) with the property that for all \( z \notin \Omega, n\left( {\gamma, z}\right) = 0 \) and for all \( z \in \Omega, n\lef... | Proof: This theorem follows from computing the residues of \( {f}^{\prime }/f \) which has residues only at poles and zeros. I will do this now. First suppose \( f \) has a pole of order \( p \) at \( \alpha \) . Then \( f \) has the form given in 53.1.3. Therefore,\n\n\[ \n\frac{{f}^{\prime }\left( z\right) }{f\left( ... | Yes |
Theorem 53.1.3 (argument principle) Let \( f \) be meromorphic in \( \Omega \) . Also suppose \( {\gamma }^{ * } \) is a closed bounded variation curve with the property that for all \( z \notin \Omega, n\left( {\gamma, z}\right) = \) 0 and for all \( z \in \Omega, n\left( {\gamma, z}\right) \) either equals 0 or 1 . N... | Proof: This theorem follows from computing the residues of \( g\left( {{f}^{\prime }/f}\right) \) . It has residues at poles and zeros. I will do this now. First suppose \( f \) has a pole of order \( m \) at \( \alpha \) . Then \( f \) has the form given in 53.1.3. Therefore,\n\n\[ g\left( z\right) \frac{{f}^{\prime }... | Yes |
Theorem 53.1.4 (Rouche’s theorem)Let \( f, g \) be meromorphic in an open set \( \Omega \) . Also suppose \( {\gamma }^{ * } \) is a closed bounded variation curve with the property that for all \( z \notin \Omega, n\left( {\gamma, z}\right) = 0 \), no zeros or poles are on \( {\gamma }^{ * } \), and for all \( z \in \... | Proof: From the hypotheses,\n\n\[ \left| {1 + \frac{f\left( z\right) }{g\left( z\right) }}\right| < 1 + \left| \frac{f\left( z\right) }{g\left( z\right) }\right| \]\n\nwhich shows that for all \( z \in {\gamma }^{ * } \) ,\n\n\[ \frac{f\left( z\right) }{g\left( z\right) } \in \mathbb{C} \smallsetminus \lbrack 0,\infty ... | Yes |
Corollary 53.1.5 In the situation of Theorem 53.1.4 change 53.1.4 to the condition, \[ \left| {f\left( z\right) - g\left( z\right) }\right| < \left| {f\left( z\right) }\right| \] for \( z \in {\gamma }^{ * } \) . Then the conclusion is the same. | Proof: The new condition implies \( \left| {1 - \frac{g}{f}\left( z\right) }\right| < \left| \frac{g\left( z\right) }{f\left( z\right) }\right| \) on \( {\gamma }^{ * } \) . Therefore, \( \frac{g\left( z\right) }{f\left( z\right) } \notin \) \( ( - \infty ,0\rbrack \) and so you can do the same argument with a branch o... | Yes |
Theorem 53.1.6 Let \( \Omega \) be a bounded open set and suppose \( f, g \) are continuous on \( \bar{\Omega } \) and analytic on \( \Omega \) . Also suppose \( \left| {f\left( z\right) }\right| < \left| {g\left( z\right) }\right| \) on \( \partial \Omega \) . Then \( g \) and \( f + g \) have the same number of zeros... | Proof: Let \( K = \{ z \in \bar{\Omega } : \left| {f\left( z\right) }\right| \geq \left| {g\left( z\right) }\right| \} \) . Then letting \( \lambda \in \left\lbrack {0,1}\right\rbrack \), if \( z \notin K \) , then \( \left| {f\left( z\right) }\right| < \left| {g\left( z\right) }\right| \) and so\n\n\[ 0 < \left| {g\le... | Yes |
Corollary 53.1.7 Let \( \\Omega \) be a bounded open set and suppose \( f, g \) are continuous on \( \\bar{\\Omega } \) and analytic on \( \\Omega \) . Also suppose \( \\left| {f\\left( z\\right) - g\\left( z\\right) }\\right| < \\left| {g\\left( z\\right) }\\right| \) on \( \\partial \\Omega \) . Then \( f \) and \( g... | Proof: You let \( f - g \) play the role of \( f \) in Theorem 53.1.6. Thus \( f - g + g = f \) and \( g \) have the same number of zeros. Alternatively, you can give a proof of this directly as follows.\n\nLet \( K = \\{ z \\in \\Omega : \\left| {f\\left( z\\right) - g\\left( z\\right) }\\right| \\geq \\left| {g\\left... | Yes |
Lemma 53.2.2 Let \( {\gamma }_{r}\left( t\right) \equiv a + r{e}^{it} \) for \( t \in \left\lbrack {0,{2\pi }}\right\rbrack \) and let \( \left| {z - a}\right| < r \) . Then \( n\left( {{\gamma }_{r}, z}\right) = 1 \) . If \( \left| {z - a}\right| > r \), then \( n\left( {{\gamma }_{r}, z}\right) = 0 \) . | Proof: For the first claim, consider for \( t \in \left\lbrack {0,1}\right\rbrack \) ,\n\n\[ f\left( t\right) \equiv n\left( {{\gamma }_{r}, a + t\left( {z - a}\right) }\right) . \]\n\nThen from properties of the winding number derived earlier, \( f\left( t\right) \in \mathbb{Z}, f \) is continuous, and \( f\left( 0\ri... | Yes |
Lemma 53.2.3 Let \( g \) be analytic on \( \operatorname{ann}\left( {a,{R}_{1},{R}_{2}}\right) \) . Then if \( {\gamma }_{r}\left( t\right) \equiv a + r{e}^{it} \) for \( t \in \left\lbrack {0,{2\pi }}\right\rbrack \) and \( r \in \left( {{R}_{1},{R}_{2}}\right) \), then \( {\int }_{{\gamma }_{r}}g\left( z\right) {dz} ... | Proof: Let \( {R}_{1} < {r}_{1} < {r}_{2} < {R}_{2} \) and denote by \( - {\gamma }_{r}\left( t\right) \) the curve, \( - {\gamma }_{r}\left( t\right) \equiv \) \( a + r{e}^{i\left( {{2\pi } - t}\right) } \) for \( t \in \left\lbrack {0,{2\pi }}\right\rbrack \) . Then if \( z \in \overline{B\left( {a,{R}_{1}}\right) } ... | Yes |
Lemma 53.2.5 Suppose\n\n\\[ \nf\\left( z\\right) = \\mathop{\\sum }\\limits_{{n = - \\infty }}^{\\infty }{a}_{n}{\\left( z - a\\right) }^{n} \n\\]\n\nfor all \\( \\left| {z - a}\\right| \\in \\left( {{R}_{1},{R}_{2}}\\right) \\) . Then both \\( \\mathop{\\sum }\\limits_{{n = 0}}^{\\infty }{a}_{n}{\\left( z - a\\right) ... | Proof: Let \\( {R}_{1} < \\left| {w - a}\\right| = {r}_{1} - \\delta < {r}_{1} \\) . Then \\( \\mathop{\\sum }\\limits_{{n = 1}}^{\\infty }{a}_{-n}{\\left( w - a\\right) }^{-n} \\) converges\n\nand so\n\n\\[ \n\\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}\\left| {a}_{-n}\\right| {\\left| w - a\\right| }^{-n} = \... | Yes |
Theorem 53.2.6 Let \( f \) be analytic on \( \operatorname{ann}\left( {a,{R}_{1},{R}_{2}}\right) \) . Then there exist numbers, \( {a}_{n} \in \mathbb{C} \) such that for all \( z \in \operatorname{ann}\left( {a,{R}_{1},{R}_{2}}\right) \) , \[ f\left( z\right) = \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{a}_{n}{... | Proof: Let \( {R}_{1} < {r}_{1} < {r}_{2} < {R}_{2} \) and define \( {\gamma }_{1}\left( t\right) \equiv a + \left( {{r}_{1} - \varepsilon }\right) {e}^{it} \) and \( {\gamma }_{2}\left( t\right) \equiv \) \( a + \left( {{r}_{2} + \varepsilon }\right) {e}^{it} \) for \( t \in \left\lbrack {0,{2\pi }}\right\rbrack \) an... | Yes |
The integral is\n\n\[ \n{\int }_{0}^{\pi }\frac{\cos \theta }{2 + \cos \theta }{d\theta } \n\] | This integrand is even and so it equals\n\n\[ \n\frac{1}{2}{\int }_{-\pi }^{\pi }\frac{\cos \theta }{2 + \cos \theta }{d\theta } \n\]\n\nFor \( z \) on the unit circle, \( z = {e}^{i\theta },\bar{z} = \frac{1}{z} \) and therefore, \( \cos \theta = \frac{1}{2}\left( {z + \frac{1}{z}}\right) \) . Thus \( {dz} = i{e}^{i\t... | Yes |
Example 53.2.10 Mellin transformations are of the form\n\n\[ \n{\int }_{0}^{\infty }f\left( x\right) {x}^{\alpha }\frac{dx}{x} \n\]\n\nSometimes it is possible to evaluate such a transform in terms of the constant, \( \alpha \) .\n\nAssume \( f \) is an analytic function except at isolated singularities, none of which ... | \n\nIn this contour the small semicircle in the center has radius \( \varepsilon \) which will converge to 0 . Denote by \( {\gamma }_{R} \) the large circular path which starts at the upper edge of the slot and co... | Yes |
Example 53.2.11 \( {\int }_{0}^{\infty }\frac{{x}^{p - 1}}{1 + x}{dx}, p \in \left( {0,1}\right) \) . | Since the exponent of \( x \) in the numerator is larger than -1 . The integral does converge. However, the techniques of real analysis don't tell us what it converges to. The contour to be used is as follows: From \( \left( {\varepsilon ,0}\right) \) to \( \left( {r,0}\right) \) along the \( x \) axis and then from \(... | Yes |
The Fresnel integrals are\n\n\[ \n{\int }_{0}^{\infty }\cos \left( {x}^{2}\right) {dx},{\int }_{0}^{\infty }\sin \left( {x}^{2}\right) {dx}.\n\] | To evaluate these integrals consider \( f\left( z\right) = {e}^{i{z}^{2}} \) on the curve which goes from the origin to the point \( r \) on the \( x \) axis and from this point to the point \( r\left( \frac{1 + i}{\sqrt{2}}\right) \) along a circle of radius \( r \), and from there back to the origin as illustrated in... | Yes |
Let \( {\gamma }_{N} \) be the contour which goes from \( - N - \frac{1}{2} - {Ni} \) horizontally to \( N + \frac{1}{2} - {Ni} \) and from there, vertically to \( N + \frac{1}{2} + {Ni} \) and then horizontally to \( - N - \frac{1}{2} + {Ni}\; \) and finally vertically to \( - N - \frac{1}{2} - {Ni} \) . Thus the cont... | You should verify that \( \cot {\pi z} \) is bounded on this contour and that therefore, \( {I}_{N} \rightarrow 0 \) as \( N \rightarrow \infty \) . Now you compute the residues of the integrand at \( \pm \alpha \) and at \( n \) where \( \left| n\right| < N + \frac{1}{2} \) for \( n \) an integer. These are the only s... | No |
Lemma 54.1.2 \( \lambda \in r\left( A\right) \) if and only if \( {\lambda I} - A \) is one to one and onto \( X \) . Also if \( \left| \lambda \right| > \parallel A\parallel \), then \( \lambda \in \sigma \left( A\right) \) . If the Neumann series,\n\n\[ \n\frac{1}{\lambda }\mathop{\sum }\limits_{{k = 0}}^{\infty }{\l... | Proof: Note that to be in \( r\left( A\right) ,{\lambda I} - A \) must be one to one and map \( X \) onto \( X \) since otherwise, \( {\left( \lambda I - A\right) }^{-1} \notin \mathcal{L}\left( {X, X}\right) \) . \n\nBy the open mapping theorem, if these two algebraic conditions hold, then \( {\left( \lambda I - A\rig... | Yes |
Lemma 54.1.3 \( r\left( A\right) \) is open. In fact, if \( \lambda \in r\left( A\right) \) and \( \left| {\mu - \lambda }\right| < {\left| \left| {\left( \lambda I - A\right) }^{-1}\right| \right| }^{-1} \) , then \( \mu \in r\left( A\right) \) . | Proof: First note\n\n\[ \left( {{\mu I} - A}\right) = \left( {I - \left( {\lambda - \mu }\right) {\left( \lambda I - A\right) }^{-1}}\right) \left( {{\lambda I} - A}\right) \]\n\n(54.1.1)\n\n\[ = \left( {{\lambda I} - A}\right) \left( {I - \left( {\lambda - \mu }\right) {\left( \lambda I - A\right) }^{-1}}\right) \]\n\... | Yes |
Corollary 54.1.4 \( \sigma \left( A\right) \) is a compact set. | Proof: Lemma 54.1.2 shows \( \sigma \left( A\right) \) is bounded and Lemma 54.1.3 shows it is closed. | Yes |
Lemma 54.1.6 If \( \left| \lambda \right| > \rho \left( A\right) \), then the Neumann series,\n\n\[ \frac{1}{\lambda }\mathop{\sum }\limits_{{k = 0}}^{\infty }{\left( \frac{A}{\lambda }\right) }^{k} \]\n\nconverges. | Proof: This follows directly from Theorem 53.2.6 on Page 1779 and the observation above that \( \frac{1}{\lambda }\mathop{\sum }\limits_{{k = 0}}^{\infty }{\left( \frac{A}{\lambda }\right) }^{k} = {\left( \lambda I - A\right) }^{-1} \) for all \( \left| \lambda \right| > \left| \right| A\left| \right| \) . Thus the ana... | Yes |
Theorem 54.1.7 \( \rho \left( A\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}{\begin{Vmatrix}{A}^{n}\end{Vmatrix}}^{1/n} \) . | Proof: If\n\n\[ \left| \lambda \right| < \lim \mathop{\sup }\limits_{{n \rightarrow \infty }}{\begin{Vmatrix}{A}^{n}\end{Vmatrix}}^{1/n} \]\nthen by the root test, the Neumann series does not converge and so by Lemma 54.1.6 \( \left| \lambda \right| \leq \rho \left( A\right) \) . Thus\n\n\[ \rho \left( A\right) \geq \l... | Yes |
Proposition 54.3.2 Suppose \( A \) is a sectorial operator as defined above so it is a densely defined closed operator on \( D\\left( A\\right) \\subseteq H \) which satisfies\n\n\[ \n\\begin{Vmatrix}{A{\\left( \\lambda I - A\\right) }^{-1}}\\end{Vmatrix} \\leq C \n\]\n\n\( \\left( {54.3.8}\\right) \)\n\nwhenever \( \\... | Proof: I need to consider \( {\\left( \\lambda I - \\left( A + B\\right) \\right) }^{-1} \) . This equals\n\n\[ \n{\\left( \\left( I - B{\\left( \\lambda I - A\\right) }^{-1}\\right) \\left( \\lambda I - A\\right) \\right) }^{-1}. \n\]\n\n(54.3.10)\n\nThe issue is whether this makes any sense for all \( \\lambda \\in {... | Yes |
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