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{"name": "Complex-valued function, single, simple pole", "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"}, "res": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a1+b1*c1^n1", "description": "", "name": "res"}, "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"}, "n1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "name": "n1"}}, "ungrouped_variables": ["a1", "n1", "c1", "b1", "res"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Complex-valued function, single, simple pole", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{c1}", "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>Pole $z_0=$ [[0]].</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{res}", "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>Corresponding residue $\\underset{z=z_0}{\\operatorname{Res}}f(z)=$ [[0]].</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{2*res}*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^{n1})/(z-{c1})},\\]</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;Single, simple pole.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a)</p>\n    <p>The given function has a simple pole at $z=\\var{c1}$.</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 $z=\\var{c1}$ is</p>\n    <p>\\[\\underset{z=\\var{c1}}{\\operatorname{Res}}=\\lim_{z\\rightarrow\\var{c1}}(z-\\var{c1})f(z)=\\lim_{z\\rightarrow\\var{c1}}\\left(\\simplify{{a1}+{b1}*z^{n1}}\\right)=\\var{res}.\\]</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\\times\\simplify{{a1}+{b1}*{c1}^{n1}}=\\simplify{2*pi*i*({a1}+{b1}*{c1}^{n1})}.\\]</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/"}]}