// Numbas version: exam_results_page_options {"name": "Principle values of log", "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(2..5)", "description": "", "name": "b1"}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(ln(a1),3)", "description": "", "name": "u"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "a1"}}, "ungrouped_variables": ["a1", "u", "b1", "tol"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Principle values of log", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "u+tol", "minValue": "u-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"answer": "pi*(1+4*n)/{2*b1}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "

Do not use decimals in your answer.

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$u=$ [[0]].  (Enter your answer to 3d.p.)

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$v=$ [[1]].  (Do not use decimals in your answer.  Use $n$ for the index, and if you need to use $\\pi$ in your answer, write pi.)

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Write the expression $\\log\\left(\\simplify{{a1}*i^(1/{b1})}\\right)$ in the form $u+iv$, where $u$ and $v$ are to be determined, and $v$ depends on some index $n=0,\\pm 1,\\pm 2,\\ldots$, say.

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15/7/2012:

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Added tags.

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Expressing $\\log(f(i))$ in the form $u+iv$.  Principal values of log.

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First write the expression $z=\\log\\left(\\simplify{{a1}*i^(1/{b1})}\\right)$ as

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\\[z=\\ln(\\var{a1})+\\log\\left(\\simplify{i^(1/{b1})}\\right),\\]

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then

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\\[z=\\ln(\\var{a1})+\\frac{1}{\\var{b1}}\\log(i).\\]

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Now

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\\[\\log(i)=\\operatorname{Log}(i)+2n\\pi i, \\quad n=0,\\pm 1,\\pm 2,\\ldots,\\]

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where $\\operatorname{Log}(i)$ is the principal value of $\\log(i)$.

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Then

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\\[\\operatorname{Log}(i)=\\ln\\lvert i\\rvert+i\\operatorname{Arg}(i)=0+\\frac{\\pi}{2}i,\\]

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so

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\\[\\log(i)=0+\\frac{\\pi}{2}i+2n\\pi i=\\frac{(1+4n)\\pi i}{2}, \\quad n=0,\\pm 1,\\pm 2,\\ldots.\\]

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Therefore

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\\[z=\\log\\left(\\simplify{{a1}*i^(1/{b1})}\\right)=\\ln(\\var{a1})+\\frac{(1+4n)\\pi i}{\\var{2*b1}}, \\quad n=0,\\pm 1,\\pm 2,\\ldots.\\]

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