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Quantum EngineeringYear 0: Mathematical FoundationsMonth 7Day 176

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Year 0·Month 7·Week 2

Day 176: Line Integrals in the Complex Plane

Day 176 of 2,016~20 min read

Learning Objectives

  • •Define and compute complex line integrals along parametrized curves
  • •Apply the fundamental theorem of calculus for complex integrals (when applicable)
  • •Understand the path-dependence of complex integrals
  • •Compute integrals using antiderivatives when they exist
  • •Relate complex line integrals to work and circulation in physics
  • •Connect to path integrals in quantum mechanics

Today's Schedule (7 hours)

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1 Complex Line Integrals DefinitionThe SettingNotation2 Basic ExamplesExample 1 Integral of z Along a Straight LineExample 2 Integral Around a CircleExample 3 The Critical Integral oint fracdzz3 Properties of Complex Line IntegralsLinearityReversalConcatenationML Inequality Estimation Lemma4 Path Independence and AntiderivativesThe Fundamental QuestionAntiderivatives inWhen Do Antiderivatives ExistExample Why oint fracdzz neq 05 Computing Integrals with AntiderivativesExample Using AntiderivativesExample Polynomial Integrals6 Integrals Involving zn Around a CircleQuantum Mechanics ConnectionPath Integrals in Quantum MechanicsAharonov-Bohm EffectBerry Phase
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