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<li><a href="/">Home</a></li>
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<li><a href="/graduate/">Graduate</a></li>
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<li><a href="/graduate/generals/">General Exams</a></li>
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<li><a href="/graduate/exams/analysis/1977-2004.html">Analysis Exams 1977-2004</a></li>
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<li aria-current="page" style="color: #666;">Complex Analysis, May 1977</li>
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<a href="/graduate/exams/analysis/1977-2004.html" class="back-button">&larr; Back to Table of Contents</a>
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<main id="main-content">
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<h2 class="unnumbered" id="complex-analysis-may-1977">Complex Analysis,
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May 1977</h2>
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<ol>
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<li><p>Classify the singularities of the following functions, including
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<math role="math" aria-label="z=\infty" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mi>&infin;</mi></mrow><annotation encoding="application/x-tex">z=\infty</annotation></semantics></math>.
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Give orders of poles.<br />
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(a)
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<math role="math" aria-label="\frac{z}{\sin z}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfrac><mi>z</mi><mrow><mrow><mi mathvariant="normal">sin</mi><mo>&#8289;</mo></mrow><mi>z</mi></mrow></mfrac><annotation encoding="application/x-tex">\frac{z}{\sin z}</annotation></semantics></math><br />
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(b)
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<math role="math" aria-label="\frac{1}{\left(1+z^{2}\right)(2-z)^{2}}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfrac><mn>1</mn><mrow><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo stretchy="false" form="prefix">(</mo><mn>2</mn><mo>−</mo><mi>z</mi><msup><mo stretchy="false" form="postfix">)</mo><mn>2</mn></msup></mrow></mfrac><annotation encoding="application/x-tex">\frac{1}{\left(1+z^{2}\right)(2-z)^{2}}</annotation></semantics></math></p></li>
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<li><p>Consider</p></li>
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</ol>
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<p><math role="math" aria-label="f(z)=\frac{1}{\left(1+z^{2}\right)(2-z)^{2}}" display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false" form="prefix">(</mo><mi>z</mi><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mrow><mo stretchy="true" form="prefix">(</mo><mn>1</mn><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup><mo stretchy="true" form="postfix">)</mo></mrow><mo stretchy="false" form="prefix">(</mo><mn>2</mn><mo>−</mo><mi>z</mi><msup><mo stretchy="false" form="postfix">)</mo><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">f(z)=\frac{1}{\left(1+z^{2}\right)(2-z)^{2}}</annotation></semantics></math></p>
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<p>(a) Find its principal part at
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<math role="math" aria-label="z=2" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">z=2</annotation></semantics></math>.<br />
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(b) In what region does its Laurant series about
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<math role="math" aria-label="z=2" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">z=2</annotation></semantics></math>
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converge?<br />
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(c) In what circular regions center at
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<math role="math" aria-label="z=0" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">z=0</annotation></semantics></math>
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does
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<math role="math" aria-label="f(z)" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false" form="prefix">(</mo><mi>z</mi><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">f(z)</annotation></semantics></math>
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have a Laurant expansion?<br />
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(d) Expand
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<math role="math" aria-label="f(z)" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false" form="prefix">(</mo><mi>z</mi><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">f(z)</annotation></semantics></math>
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in
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<math role="math" aria-label="1&lt;|z|&lt;2" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>&lt;</mo><mo stretchy="false" form="prefix">|</mo><mi>z</mi><mo stretchy="false" form="prefix">|</mo><mo>&lt;</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">1&lt;|z|&lt;2</annotation></semantics></math>.<br />
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3. Integrate, by complex methods,<br />
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(a)
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<math role="math" aria-label="\int_{-\pi}^{\pi} \frac{1}{1-a \cos \theta} d \theta, \quad 0&lt;a&lt;1" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>&int;</mo><mrow><mi>−</mi><mi>&pi;</mi></mrow><mi>&pi;</mi></msubsup><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><mi>a</mi><mrow><mi mathvariant="normal">cos</mi><mo>&#8289;</mo></mrow><mi>&theta;</mi></mrow></mfrac><mi>d</mi><mi>&theta;</mi><mo>,</mo><mspace width="1.0em"></mspace><mn>0</mn><mo>&lt;</mo><mi>a</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\int_{-\pi}^{\pi} \frac{1}{1-a \cos \theta} d \theta, \quad 0&lt;a&lt;1</annotation></semantics></math>;<br />
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(b)
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<math role="math" aria-label="\int_{0}^{\infty} \frac{d x}{x^{\alpha}(x+1)^{2}}, \quad 0&lt;\alpha&lt;1" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>&int;</mo><mn>0</mn><mi>&infin;</mi></msubsup><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><msup><mi>x</mi><mi>&alpha;</mi></msup><mo stretchy="false" form="prefix">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><msup><mo stretchy="false" form="postfix">)</mo><mn>2</mn></msup></mrow></mfrac><mo>,</mo><mspace width="1.0em"></mspace><mn>0</mn><mo>&lt;</mo><mi>&alpha;</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\int_{0}^{\infty} \frac{d x}{x^{\alpha}(x+1)^{2}}, \quad 0&lt;\alpha&lt;1</annotation></semantics></math>.<br />
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4. Find, if possible, a conformal map of
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<math role="math" aria-label="A" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>A</mi><annotation encoding="application/x-tex">A</annotation></semantics></math>
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onto the upper half plane
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<math role="math" aria-label="\{z: \operatorname{Im} z&gt;0\}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false" form="prefix">{</mo><mi>z</mi><mo>:</mo><mrow><mi mathvariant="normal">Im</mi><mo>&#8289;</mo></mrow><mi>z</mi><mo>&gt;</mo><mn>0</mn><mo stretchy="false" form="postfix">}</mo></mrow><annotation encoding="application/x-tex">\{z: \operatorname{Im} z&gt;0\}</annotation></semantics></math>,
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where<br />
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(a)
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<math role="math" aria-label="A" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>A</mi><annotation encoding="application/x-tex">A</annotation></semantics></math>
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is the complex plane;<br />
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(b)
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<math role="math" aria-label="A" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>A</mi><annotation encoding="application/x-tex">A</annotation></semantics></math>
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is the unit disc
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<math role="math" aria-label="\{z:|z|&lt;1\}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false" form="prefix">{</mo><mi>z</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">\{z:|z|&lt;1\}</annotation></semantics></math>.</p>
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