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Quantum Engineering Year 0: Mathematical Foundations Month 7Day 179

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Year 0·Month 7·Week 2

Day 179: Applications to Real Integrals I

Day 179 of 2,016~13 min read

Learning Objectives

•Evaluate improper integrals using semicircular contours
•Apply the residue-like technique for simple poles
•Handle integrals of rational functions
•Understand Jordan's lemma for exponential integrands
•Compute Fourier-type integrals
•Connect to quantum scattering amplitudes

Today's Schedule (7 hours)

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1 The Strategy 2 Integrals of Rational Functions Type I int_-inftyinfty fracPxQx dx The Semicircular Contour 3 Computing Residues Preview 4 Example int_-inftyinfty fracdx1x2 5 Example int_-inftyinfty fracdx1x22 6 Jordans Lemma 7 Fourier-Type Integrals Type II int_-inftyinfty fx eiax dx a 0 Example int_-inftyinfty fraceix1x2 dx Extracting Real Parts 8 Integrals with Branch Points Quantum Mechanics Connection Scattering Amplitudes Density of States Kramers-Kronig Relations

Day 178Day 179 of 2,016Day 180