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EXAM SPRING 2022 - UVA Mathematics</title>
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<li><a href="/">Home</a></li>
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<li><a href="/graduate/generals/">General Exams</a></li>
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<li aria-current="page" style="color: #666;">COMPLEX ANALYSIS GENERAL
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EXAM SPRING 2022</li>
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<a href="/graduate/generals/" style="display: inline-block; padding: 0.5em 1em; background-color: #f0f0f0; border: 1px solid #ccc; border-radius: 4px; text-decoration: none; color: #333;">&larr; Back to General Exams</a>
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<main>
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<h1 class="unnumbered"
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id="complex-analysis-general-exam-spring-2022">COMPLEX ANALYSIS GENERAL
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EXAM SPRING 2022</h1>
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<p>Solve as many problems as you can. Full solutions on a smaller number
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of problems will be worth more than partial solutions on several
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problems. Throughout
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<math role="math" aria-label="\mathbb{D}=\{z \in \mathbb{C}:|z|&lt;1\}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>𝔻</mi><mo>=</mo><mo stretchy="false" form="prefix">{</mo><mi>z</mi><mo>&isin;</mo><mi>&Copf;</mi><mo>:</mo><mo stretchy="false" form="prefix">|</mo><mi>z</mi><mo stretchy="false" form="prefix">|</mo><mo>&lt;</mo><mn>1</mn><mo stretchy="false" form="postfix">}</mo></mrow><annotation encoding="application/x-tex">\mathbb{D}=\{z \in \mathbb{C}:|z|&lt;1\}</annotation></semantics></math>.
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Don’t use any of the Picard theorems.</p>
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<h2 class="unnumbered" id="problem-1">Problem 1</h2>
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<p>Compute, for
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<math role="math" aria-label="0&lt;\alpha&lt;1" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>&lt;</mo><mi>&alpha;</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0&lt;\alpha&lt;1</annotation></semantics></math></p>
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<p><math role="math" aria-label="\int_{0}^{\infty} \frac{1}{x^{\alpha}(1+x)} d x" display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>&int;</mo><mn>0</mn><mi>&infin;</mi></msubsup><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mi>&alpha;</mi></msup><mo stretchy="false" form="prefix">(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo stretchy="false" form="postfix">)</mo></mrow></mfrac><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\int_{0}^{\infty} \frac{1}{x^{\alpha}(1+x)} d x</annotation></semantics></math></p>
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<p>Show all estimates.</p>
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<h2 class="unnumbered" id="problem-2">Problem 2</h2>
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<p>Suppose that
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<math role="math" aria-label="f" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>f</mi><annotation encoding="application/x-tex">f</annotation></semantics></math>
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is an entire function, and that there are constants
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<math role="math" aria-label="a, b&gt;0" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a, b&gt;0</annotation></semantics></math>
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so that</p>
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<p><math role="math" aria-label="|f(z)| \leq a+b|z| \text { for all } z \in \mathbb{C}" display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false" form="prefix">|</mo><mi>f</mi><mo stretchy="false" form="prefix">(</mo><mi>z</mi><mo stretchy="false" form="postfix">)</mo><mo stretchy="false" form="prefix">|</mo><mo>&le;</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false" form="prefix">|</mo><mi>z</mi><mo stretchy="false" form="prefix">|</mo><mrow><mspace width="0.333em"></mspace><mtext mathvariant="normal"> for all </mtext><mspace width="0.333em"></mspace></mrow><mi>z</mi><mo>&isin;</mo><mi>&Copf;</mi></mrow><annotation encoding="application/x-tex">|f(z)| \leq a+b|z| \text { for all } z \in \mathbb{C}</annotation></semantics></math></p>
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<p>Show that
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<math role="math" aria-label="f" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>f</mi><annotation encoding="application/x-tex">f</annotation></semantics></math>
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is a polynomial of degree at most one.</p>
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<h2 class="unnumbered" id="problem-3">Problem 3</h2>
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<p>Give an explicit example of an unbounded harmonic function
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<math role="math" aria-label="u: \mathbb{D} \rightarrow(0,+\infty)" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>:</mo><mi>𝔻</mi><mo>&rarr;</mo><mo stretchy="false" form="prefix">(</mo><mn>0</mn><mo>,</mo><mi>+</mi><mi>&infin;</mi><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">u: \mathbb{D} \rightarrow(0,+\infty)</annotation></semantics></math>
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with the property that</p>
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<p><math role="math" aria-label="\lim _{z \rightarrow \zeta} u(z)=0" display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder><mi mathvariant="normal">lim</mi><mrow><mi>z</mi><mo>&rarr;</mo><mi>&zeta;</mi></mrow></munder><mi>u</mi><mo stretchy="false" form="prefix">(</mo><mi>z</mi><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lim _{z \rightarrow \zeta} u(z)=0</annotation></semantics></math></p>
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<p>for all
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<math role="math" aria-label="\zeta \in \partial \mathbb{D}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>&zeta;</mi><mo>&isin;</mo><mi>&part;</mi><mi>𝔻</mi></mrow><annotation encoding="application/x-tex">\zeta \in \partial \mathbb{D}</annotation></semantics></math>
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with
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<math role="math" aria-label="\zeta \neq 1" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>&zeta;</mi><mo>&ne;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\zeta \neq 1</annotation></semantics></math>.
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It is acceptable to leave your answer as the real (or imaginary) part of
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an explicit holomorphic function.</p>
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<h2 class="unnumbered" id="problem-4">Problem 4</h2>
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<p>Let
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<math role="math" aria-label="S=\{z \in \mathbb{C}: 0&lt;\operatorname{Re}(z)&lt;1\}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo>=</mo><mo stretchy="false" form="prefix">{</mo><mi>z</mi><mo>&isin;</mo><mi>&Copf;</mi><mo>:</mo><mn>0</mn><mo>&lt;</mo><mrow><mi mathvariant="normal">Re</mi><mo>&#8289;</mo></mrow><mo stretchy="false" form="prefix">(</mo><mi>z</mi><mo stretchy="false" form="postfix">)</mo><mo>&lt;</mo><mn>1</mn><mo stretchy="false" form="postfix">}</mo></mrow><annotation encoding="application/x-tex">S=\{z \in \mathbb{C}: 0&lt;\operatorname{Re}(z)&lt;1\}</annotation></semantics></math>.
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Suppose that
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<math role="math" aria-label="f: \bar{S} \rightarrow \mathbb{C}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>:</mo><mover><mi>S</mi><mo accent="true">‾</mo></mover><mo>&rarr;</mo><mi>&Copf;</mi></mrow><annotation encoding="application/x-tex">f: \bar{S} \rightarrow \mathbb{C}</annotation></semantics></math>
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is bounded, continuous, and that
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<math role="math" aria-label="\left.f\right|_{S}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mrow><mi>f</mi><mo stretchy="true" form="postfix">|</mo></mrow><mi>S</mi></msub><annotation encoding="application/x-tex">\left.f\right|_{S}</annotation></semantics></math>
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is analytic. If</p>
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<p><math role="math" aria-label="\sup _{t \in \mathbb{R}} \max (|f(i t)|,|f(1+i t)|) \leq 1," display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder><mi mathvariant="normal">sup</mi><mrow><mi>t</mi><mo>&isin;</mo><mi>&Ropf;</mi></mrow></munder><mrow><mi mathvariant="normal">max</mi><mo>&#8289;</mo></mrow><mo stretchy="false" form="prefix">(</mo><mo stretchy="false" form="prefix">|</mo><mi>f</mi><mo stretchy="false" form="prefix">(</mo><mi>i</mi><mi>t</mi><mo stretchy="false" form="postfix">)</mo><mo stretchy="false" form="prefix">|</mo><mo>,</mo><mo stretchy="false" form="prefix">|</mo><mi>f</mi><mo stretchy="false" form="prefix">(</mo><mn>1</mn><mo>+</mo><mi>i</mi><mi>t</mi><mo stretchy="false" form="postfix">)</mo><mo stretchy="false" form="prefix">|</mo><mo stretchy="false" form="postfix">)</mo><mo>&le;</mo><mn>1</mn><mo>,</mo></mrow><annotation encoding="application/x-tex">\sup _{t \in \mathbb{R}} \max (|f(i t)|,|f(1+i t)|) \leq 1,</annotation></semantics></math></p>
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<p>show that
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<math role="math" aria-label="|f(z)| \leq 1" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false" form="prefix">|</mo><mi>f</mi><mo stretchy="false" form="prefix">(</mo><mi>z</mi><mo stretchy="false" form="postfix">)</mo><mo stretchy="false" form="prefix">|</mo><mo>&le;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|f(z)| \leq 1</annotation></semantics></math>
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for all
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<math role="math" aria-label="z \in S" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>&isin;</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">z \in S</annotation></semantics></math>.<br />
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Suggestion: consider, for
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<math role="math" aria-label="\varepsilon&gt;0" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>&epsilon;</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\varepsilon&gt;0</annotation></semantics></math>,
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the function
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<math role="math" aria-label="f_{\varepsilon}(z)=\frac{f(z)}{1+\varepsilon z}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>&epsilon;</mi></msub><mo stretchy="false" form="prefix">(</mo><mi>z</mi><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mfrac><mrow><mi>f</mi><mo stretchy="false" form="prefix">(</mo><mi>z</mi><mo stretchy="false" form="postfix">)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mi>&epsilon;</mi><mi>z</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">f_{\varepsilon}(z)=\frac{f(z)}{1+\varepsilon z}</annotation></semantics></math>.
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Show that
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<math role="math" aria-label="\left|f_{\varepsilon}\right| \leq 1" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo stretchy="true" form="prefix">|</mo><msub><mi>f</mi><mi>&epsilon;</mi></msub><mo stretchy="true" form="postfix">|</mo></mrow><mo>&le;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\left|f_{\varepsilon}\right| \leq 1</annotation></semantics></math>
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for all
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<math role="math" aria-label="\varepsilon&gt;0" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>&epsilon;</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\varepsilon&gt;0</annotation></semantics></math>,
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and use this to conclude that
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<math role="math" aria-label="|f| \leq 1" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false" form="prefix">|</mo><mi>f</mi><mo stretchy="false" form="prefix">|</mo><mo>&le;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|f| \leq 1</annotation></semantics></math>.</p>
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<h2 class="unnumbered" id="problem-5">Problem 5</h2>
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<p>Let
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<math role="math" aria-label="\left(M_{n}\right)_{n=0}^{\infty}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mrow><mo stretchy="true" form="prefix">(</mo><msub><mi>M</mi><mi>n</mi></msub><mo stretchy="true" form="postfix">)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mi>&infin;</mi></msubsup><annotation encoding="application/x-tex">\left(M_{n}\right)_{n=0}^{\infty}</annotation></semantics></math>
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be a sequence of positive real numbers. Assume that the series
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<math role="math" aria-label="\sum_{n=0}^{\infty} M_{n} z^{n}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>&sum;</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mi>&infin;</mi></msubsup><msub><mi>M</mi><mi>n</mi></msub><msup><mi>z</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\sum_{n=0}^{\infty} M_{n} z^{n}</annotation></semantics></math>
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has radius of convergence at least 1 . Let
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<math role="math" aria-label="\mathcal{F}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>ℱ</mi><annotation encoding="application/x-tex">\mathcal{F}</annotation></semantics></math>
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be the set of holomorphic functions on
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<math role="math" aria-label="\mathbb{D}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>𝔻</mi><annotation encoding="application/x-tex">\mathbb{D}</annotation></semantics></math>
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which satisfy</p>
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<p><math role="math" aria-label="\left|\frac{f^{(n)}(0)}{n!}\right| \leq M_{n} \text { for all } n \in \mathbb{N} \cup\{0\} ." display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo stretchy="true" form="prefix">|</mo><mfrac><mrow><msup><mi>f</mi><mrow><mo stretchy="false" form="prefix">(</mo><mi>n</mi><mo stretchy="false" form="postfix">)</mo></mrow></msup><mo stretchy="false" form="prefix">(</mo><mn>0</mn><mo stretchy="false" form="postfix">)</mo></mrow><mrow><mi>n</mi><mi>!</mi></mrow></mfrac><mo stretchy="true" form="postfix">|</mo></mrow><mo>&le;</mo><msub><mi>M</mi><mi>n</mi></msub><mrow><mspace width="0.333em"></mspace><mtext mathvariant="normal"> for all </mtext><mspace width="0.333em"></mspace></mrow><mi>n</mi><mo>&isin;</mo><mi>&Nopf;</mi><mo>&cup;</mo><mo stretchy="false" form="prefix">{</mo><mn>0</mn><mo stretchy="false" form="postfix">}</mo><mi>.</mi></mrow><annotation encoding="application/x-tex">\left|\frac{f^{(n)}(0)}{n!}\right| \leq M_{n} \text { for all } n \in \mathbb{N} \cup\{0\} .</annotation></semantics></math></p>
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<p>Show that
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<math role="math" aria-label="\mathcal{F}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>ℱ</mi><annotation encoding="application/x-tex">\mathcal{F}</annotation></semantics></math>
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is normal (i.e. the closure of
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<math role="math" aria-label="\mathcal{F}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>ℱ</mi><annotation encoding="application/x-tex">\mathcal{F}</annotation></semantics></math>
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is compact for the topology of uniform convergence on compact subsets of
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<math role="math" aria-label="\mathbb{D}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>𝔻</mi><annotation encoding="application/x-tex">\mathbb{D}</annotation></semantics></math>
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).</p>
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<p>? =0 <a href="#fn1" class="footnote-ref" id="fnref1"
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role="doc-noteref"><sup>1</sup></a></p>
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<p>??=0 <a href="#fn2" class="footnote-ref" id="fnref2"
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role="doc-noteref"><sup>2</sup></a></p>
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<section id="footnotes" class="footnotes footnotes-end-of-document"
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<hr />
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<ol>
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<li id="fn1"><p>Date: January 12, 2022.<a href="#fnref1"
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class="footnote-back" role="doc-backlink">↩︎</a></p></li>
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<li id="fn2"><p>Date: January 12, 2022.<a href="#fnref2"
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class="footnote-back" role="doc-backlink">↩︎</a></p></li>
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</section>
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