Methods of Complex Functions

methods-of-complex-functions

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355584

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What is the Anti-Derivative Theorem?

Let f be holomorphic on a simply connected domain S. Then there exists a function F, holomorphic on S, with F' = f

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355586

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What is Liouville's Theorem?

Any bounded entire function is constant

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{"id":1595321,"question_id":"355586","created_at":"2018-01-09 15:26:21","updated_at":"2018-01-09 15:26:21","answerTitle":"Any bounded entire function is constant","isAnswerCorrect":"1","isAnswerBlank":null,"isInSecondMatchingColumn":null,"sequenceOrder":null,"lang":null,"type":0}

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355590

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How do you characterise poles?

A pole is of order m if Lim (z-a)^mf(z) = D

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{"id":1595325,"question_id":"355590","created_at":"2018-01-09 15:29:24","updated_at":"2018-01-09 15:29:24","answerTitle":"A pole is of order m if Lim (z-a)^mf(z) = D","isAnswerCorrect":"1","isAnswerBlank":null,"isInSecondMatchingColumn":null,"sequenceOrder":null,"lang":null,"type":0}

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355592

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What is Cauchy's Residue Theorem?

Let f be holomorphic on and inside a simple, positively oriented curve Y except at a finite number of poles inside Y. Then..

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{"id":1595327,"question_id":"355592","created_at":"2018-01-09 15:30:26","updated_at":"2018-01-09 15:30:26","answerTitle":"Let f be holomorphic on and inside a simple, positively oriented curve Y except at a finite number of poles inside Y. Then..","isAnswerCorrect":"1","isAnswerBlank":null,"isInSecondMatchingColumn":null,"sequenceOrder":null,"lang":null,"type":0}

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355552

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What is an open set?

a set S is open if there is some r > 0 such that for any z in S there exists a subset D(z;r) in S

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{"id":1595280,"question_id":"355552","created_at":"2018-01-09 14:30:07","updated_at":"2018-01-09 14:30:07","answerTitle":"a set S is open if there is some r > 0 such that for any z in S there exists a subset D(z;r) in S","isAnswerCorrect":"1","isAnswerBlank":null,"isInSecondMatchingColumn":null,"sequenceOrder":null,"lang":null,"type":0}

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355576

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What is a connected set?

A set is connected if each pair of points can be connected by a curve which also lies in S

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{"id":1595310,"question_id":"355576","created_at":"2018-01-09 15:17:07","updated_at":"2018-01-09 15:17:07","answerTitle":"A set is connected if each pair of points can be connected by a curve which also lies in S","isAnswerCorrect":"1","isAnswerBlank":null,"isInSecondMatchingColumn":null,"sequenceOrder":null,"lang":null,"type":0}

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355577

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What is a region?

Open, connected, non-empty set

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{"id":1595311,"question_id":"355577","created_at":"2018-01-09 15:17:33","updated_at":"2018-01-09 15:17:33","answerTitle":"Open, connected, non-empty set","isAnswerCorrect":"1","isAnswerBlank":null,"isInSecondMatchingColumn":null,"sequenceOrder":null,"lang":null,"type":0}

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355579

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What does simply connected mean?

Let S be a region. S is simply connected if for any closed curve, the interior is wholly in S

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{"id":1595313,"question_id":"355579","created_at":"2018-01-09 15:18:35","updated_at":"2018-01-09 15:18:35","answerTitle":"Let S be a region. S is simply connected if for any closed curve, the interior is wholly in S","isAnswerCorrect":"1","isAnswerBlank":null,"isInSecondMatchingColumn":null,"sequenceOrder":null,"lang":null,"type":0}

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355580

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What is Cauchy's Theorem?

Let f be holomorphic in a simply connected domain S, then the integral of f(z)dz = o for every closed path in S

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{"id":1595314,"question_id":"355580","created_at":"2018-01-09 15:20:27","updated_at":"2018-01-09 15:20:27","answerTitle":"Let f be holomorphic in a simply connected domain S, then the integral of f(z)dz = o for every closed path in S","isAnswerCorrect":"1","isAnswerBlank":null,"isInSecondMatchingColumn":null,"sequenceOrder":null,"lang":null,"type":0}

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355582

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What is Cauchy's Integral Formula for derivatives?

Let f be holomorphic in Y, and a be inside Y. Then f^(n)(a) = n!/2pii(integral of f(z)/(z-a)^n+1 dz

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{"id":1595316,"question_id":"355582","created_at":"2018-01-09 15:22:51","updated_at":"2018-01-09 15:22:51","answerTitle":"Let f be holomorphic in Y, and a be inside Y. Then f^(n)(a) = n!\/2pii(integral of f(z)\/(z-a)^n+1 dz","isAnswerCorrect":"1","isAnswerBlank":null,"isInSecondMatchingColumn":null,"sequenceOrder":null,"lang":null,"type":0}

Methods of Complex Functions

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