// Numbas version: exam_results_page_options
{"name": "Complex-valued function, pair of pure imaginary 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(1..4)", "description": "", "name": "b1"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "a1"}}, "ungrouped_variables": ["a1", "b1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Complex-valued function, pair of pure imaginary poles", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{b1}*i", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{-b1}*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>There are two poles $z_0$ and $z_1$. &nbsp;Enter the pole with the <strong>greatest imaginary part</strong> first.</p>\n        <p>$z_0=$ [[0]].</p>\n        <p>$z_1=$ [[1]].</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "-i*{cos(a1*b1*pi)}/{2*b1}", "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": "i*{cos(a1*b1*pi)}/{2*b1}", "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": "0", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Do not use the exponential function, or any trigonometric functions in your answer.</p>", "showStrings": false, "partialCredit": 0, "strings": ["e", "exp", "cos", "sin", "tan", "sec", "cosec", "cot"]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>$I=$ [[0]].</p>\n        <p>Do not use the exponential function, or any trigonometric functions in your answer.</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>For the function</p>\n    <p>\\[f(z)=\\frac{\\mathrm{e}^{\\simplify{{a1}*pi*z}}}{\\simplify{z^2+{b1^2}}},\\]</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 pure imaginary poles.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a)</p>\n    <p>The given function has two simple poles at $z=\\simplify{{b1}*i}$ and $z=\\simplify{{-b1}*i}$.</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 $z=\\simplify{{b1}*i}$ is</p>\n    <p>\\[\\underset{z=\\simplify{{b1}*i}}{\\operatorname{Res}}=\\lim_{z\\rightarrow\\simplify{{b1}*i}}(z-\\simplify{{b1}*i})f(z)=\\lim_{z\\rightarrow\\simplify{{b1}*i}}\\left(\\frac{\\mathrm{e}^{\\simplify{{a1}*pi*z}}}{\\simplify{z+{b1}*i}}\\right)=\\simplify{-i/{2*b1}}\\mathrm{e}^{\\simplify{{a1*b1}*pi*i}}=\\frac{1}{\\var{2*b1}}\\biggl(\\sin(\\simplify{{a1*b1}*pi})-i\\cos(\\simplify{{a1*b1}*pi})\\biggr)=\\simplify{-i*cos({a1*b1*pi})/{2*b1}},\\]</p>\n    <p>and the corresponding residue for $z=\\simplify{{-b1}*i}$ is</p>\n    <p>\\[\\underset{z=\\simplify{{-b1}*i}}{\\operatorname{Res}}=\\lim_{z\\rightarrow\\simplify{{-b1}*i}}(z+\\simplify{{b1}*i})f(z)=\\lim_{z\\rightarrow\\simplify{{-b1}*i}}\\left(\\frac{\\mathrm{e}^{\\simplify{{a1}*pi*z}}}{\\simplify{z-{b1}*i}}\\right)=\\simplify{i/{2*b1}}\\mathrm{e}^{\\simplify{{-a1*b1*pi*i}}}=\\frac{1}{\\var{2*b1}}\\biggl(\\sin(\\simplify{{a1*b1}*pi})+i\\cos(\\simplify{{a1*b1}*pi})\\biggr)=\\simplify{i*cos({a1*b1*pi})/{2*b1}}.\\]</p>\n    <p>&nbsp;</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{i*cos({a1*b1*pi})/{2*b1}}-\\simplify{i*cos({a1*b1*pi})/{2*b1}}\\right)=0.\\]</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/"}]}