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Quantum Engineering Year 0: Mathematical Foundations Month 7Day 178

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Year 0·Month 7·Week 2

Day 178: Cauchy's Integral Formula

Day 178 of 2,016~16 min read

Learning Objectives

•State and prove Cauchy's integral formula
•Derive the formula for derivatives of analytic functions
•Apply the formulas to compute values and derivatives
•Understand the maximum modulus principle
•Prove Liouville's theorem and the fundamental theorem of algebra
•Connect to quantum mechanical propagators

Today's Schedule (7 hours)

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1 Cauchys Integral Formula The Central Result Proof 2 Formula for Derivatives First Derivative Higher Derivatives 3 Moreras Theorem Converse of Cauchy 4 Cauchys Inequality 5 Liouvilles Theorem 6 Fundamental Theorem of Algebra 7 Maximum Modulus Principle Quantum Mechanics Connection Propagators and Greens Functions Spectral Decomposition Analyticity and Causality

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