// Numbas version: exam_results_page_options
{"name": "Complex-valued function, pair of real poles", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "name": "b1"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "c1"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "a1"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "d1"}}, "ungrouped_variables": ["a1", "c1", "b1", "d1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Complex-valued function, pair of real poles", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "0", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{d1}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>There are two poles $z_0$ and $z_1$. &nbsp;Enter the pole with the <strong>least&nbsp;real part</strong>&nbsp;first.</p>\n        <p>$z_0=$ [[0]].</p>\n        <p>$z_1=$ [[1]].</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "-{a1}/{d1}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}, {"answer": "{a1+b1*d1+c1*d1^2}/{d1}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>Corresponding residue $\\underset{z=z_0}{\\operatorname{Res}}f(z)=$ [[0]].</p>\n        <p>Corresponding residue $\\underset{z=z_1}{\\operatorname{Res}}f(z)=$ [[1]].</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{2*(b1+c1*d1)*pi*i}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>$I=$ [[0]].</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>For the function</p>\n    <p>\\[f(z)=\\simplify{({a1}+{b1}*z+{c1}*z^2)/(z^2-{d1}*z)},\\]</p>\n    <p>identify the poles (singular points) $z_0$, the corresponding residues $\\underset{z=z_0}{\\operatorname{Res}}f(z)$, and evaluate the integral</p>\n    <p>\\[I=\\oint{\\!f(z)\\,\\mathrm{d}z},\\]</p>\n    <p>where $C$ is the contour $\\lvert z \\rvert=5$ mapped counter-clockwise.</p>", "tags": ["checked2015", "MAS2103"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>15/7/2012:</strong></p>\n<p>Added tags.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Poles, residues, and contour integral of a complex-valued function. &nbsp;Pair of real poles.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a)</p>\n    <p>The given function has two simple poles at $z=0$ and $z=\\var{d1}$.</p>\n    <p>&nbsp;</p>\n    <p>b)</p>\n    <p>For a simple pole $z=z_0$, the residue can be calculated using the formula</p>\n    <p>\\[\\underset{z=z_0}{\\operatorname{Res}}=\\lim_{z\\rightarrow z_0}(z-z_0)f(z).\\]</p>\n    <p>In this case, the corresponding residue for $z=0$ is</p>\n    <p>\\[\\underset{z=0}{\\operatorname{Res}}=\\lim_{z\\rightarrow 0}zf(z)=\\lim_{z\\rightarrow0}\\left(\\simplify{({a1}+{b1}*z+{c1}*z^2)/(z-{d1})}\\right)=\\simplify{-{a1}/{d1}}.\\]</p>\n    <p>and the corresponding residue for $z=\\var{d1}$ is</p>\n    <p>\\[\\underset{z=\\var{d1}}{\\operatorname{Res}}=\\lim_{z\\rightarrow\\var{d1}}(z-\\var{d1})f(z)=\\lim_{z\\rightarrow\\var{d1}}\\left(\\simplify{({a1}+{b1}*z+{c1}*z^2)/z}\\right)=\\simplify{{a1+b1*d1+c1*d1^2}/{d1}}.\\]</p>\n    <p>c)</p>\n    <p>Now use the Residue Theorem to calculate the integral $I$.</p>\n    <h4>Residue Theorem</h4>\n    <blockquote>Let $f(z)$ be analytic inside a closed path $C$, and on $C$, except at finitely many singular points $z_n=z_1,z_2,\\ldots,z_k$ inside $C$. &nbsp;Then the integral of $f(z)$ taken counter-clockwise around $C$ equals $2\\pi i$ times the sum of the residues of $f(z)$ at $z_n=z_1,z_2,\\ldots,z_k$, i.e. \\[\\oint_C{\\!f(z)\\,\\mathrm{d}z}=2\\pi i\\sum_{n=1}^k{\\underset{z=z_n}{\\operatorname{Res}}f(z)}.\\]</blockquote>\n    <p>Therefore</p>\n    <p>\\[I=2\\pi i\\left(\\simplify{-{a1}/{d1}}+\\simplify{{a1+b1*d1+c1*d1^2}/{d1}}\\right)=\\simplify{{2*pi*i*(b1+c1*d1)}}.\\]</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}