# Day 140: Week 20 Review — Complex Analysis II Mastery ## 📅 Schedule Overview | Block | Time | Duration | Activity | |-------|------|----------|----------| | Morning | 9:00 AM - 12:00 PM | 3 hours | Concept Review & Problem Set A | | Afternoon | 2:00 PM - 5:00 PM | 3 hours | Problem Set B & Self-Assessment | | Evening | 6:00 PM - 7:00 PM | 1 hour | Month 6 Preview & Planning | **Total Study Time: 7 hours** --- ## 🎯 Week 20 Learning Objectives — Final Check By the end of this review, confirm mastery of: - [ ] Laurent series expansions - [ ] Classification of singularities (removable, poles, essential) - [ ] Residue computation methods - [ ] The Residue Theorem - [ ] Evaluation of real integrals via residues - [ ] Argument Principle and winding numbers - [ ] Rouché's Theorem and zero counting - [ ] Physics applications (Green's functions, scattering) --- ## 📊 Week 20 Concept Map ``` COMPLEX ANALYSIS II │ ┌──────────────────┼──────────────────┐ │ │ │ SINGULARITIES RESIDUES APPLICATIONS │ │ │ ┌────┴────┐ ┌────┴────┐ ┌────┴────┐ │ │ │ │ │ │ Laurent Classification Residue Real Argument Series │ Theorem Integrals Principle │ ┌────┴────┐ │ │ │ │ │ │ │ │ ┌────┴────┐ │ Principal Remov Poles Ess ∮=2πi Rational Winding Part │ │ ΣRes Trig Numbers │ │ │ │ Fourier │ │ f bounded |f|→∞ Wild │ Rouché │ │ │ behavior │ Theorem Residue Res=0 Res= Infinite │ │ = a₋₁ formula series │ Zero Jordan Counting Lemma ``` --- ## 📝 Key Formulas Summary ### Laurent Series $$f(z) = \sum_{n=-\infty}^{\infty} a_n(z-z_0)^n, \quad a_n = \frac{1}{2\pi i}\oint_C \frac{f(z)}{(z-z_0)^{n+1}}dz$$ ### Singularity Classification | Type | Principal Part | Behavior | |------|---------------|----------| | Removable | None | f bounded | | Pole (order m) | Finite (m terms) | \|f\| → ∞ | | Essential | Infinite | Casorati-Weierstrass | ### Residue Formulas | Case | Formula | |------|---------| | Simple pole | Res = lim_{z→z₀} (z-z₀)f(z) | | Quotient g/h | Res = g(z₀)/h'(z₀) | | Pole order m | Res = (1/(m-1)!) lim d^{m-1}/dz^{m-1}[(z-z₀)^m f(z)] | ### Residue Theorem $$\oint_C f(z)\,dz = 2\pi i \sum_{\text{inside}} \text{Res}[f, z_k]$$ ### Real Integrals | Type | Method | |------|--------| | ∫ p(x)/q(x) dx | Semicircle, upper poles | | ∫ R(cos,sin) dθ | Unit circle, z = e^{iθ} | | ∫ f(x)e^{iax} dx | Jordan's lemma | ### Argument Principle $$\frac{1}{2\pi i}\oint_C \frac{f'(z)}{f(z)}dz = Z - P$$ ### Rouché's Theorem If \|g(z)\| < \|f(z)\| on C, then f and f+g have the same number of zeros inside C. --- ## 🔬 Problem Set A: Laurent Series & Singularities ### Problem A1: Laurent Series Find the Laurent series for f(z) = 1/((z-1)(z-2)) in: a) |z| < 1 b) 1 < |z| < 2 c) |z| > 2 **Solutions:** a) Both poles outside, Taylor series: $$f(z) = \frac{1}{(z-1)(z-2)} = \frac{-1}{z-1} + \frac{1}{z-2} = \frac{1}{1-z} - \frac{1}{2(1-z/2)}$$ $$= \sum_{n=0}^{\infty} z^n - \frac{1}{2}\sum_{n=0}^{\infty}\frac{z^n}{2^n} = \sum_{n=0}^{\infty}\left(1 - \frac{1}{2^{n+1}}\right)z^n$$ b) Pole at z=1 inside, pole at z=2 outside: $$\frac{1}{z-1} = \frac{1}{z(1-1/z)} = \frac{1}{z}\sum_{n=0}^{\infty}\frac{1}{z^n}$$ $$f(z) = -\sum_{n=0}^{\infty}z^{-n-1} - \frac{1}{2}\sum_{n=0}^{\infty}\frac{z^n}{2^n}$$ c) Both poles inside relative to ∞: $$f(z) = -\sum_{n=0}^{\infty}z^{-n-1} + \sum_{n=0}^{\infty}\frac{2^n}{z^{n+1}}$$ --- ### Problem A2: Classify Singularities Classify the singularity at z = 0: a) f(z) = sin z/z b) f(z) = (1 - cos z)/z² c) f(z) = e^{1/z²} d) f(z) = z²/(e^z - 1) **Solutions:** a) sin z/z = 1 - z²/6 + ... → **Removable** (define f(0) = 1) b) (1 - cos z)/z² = (z²/2 - z⁴/24 + ...)/z² = 1/2 - z²/24 + ... → **Removable** (f(0) = 1/2) c) e^{1/z²} = 1 + 1/z² + 1/(2z⁴) + ... → **Essential** (infinitely many negative powers) d) e^z - 1 = z + z²/2 + ..., so z²/(e^z-1) = z/(1 + z/2 + ...) = z - z²/2 + ... → **Removable** at z=0 --- ### Problem A3: Residue Computation Find all residues of f(z) = z/((z²+1)(z-2)): **Solution:** Poles at z = i, -i, 2 (all simple). At z = i: Res = i/((2i)(i-2)) = i/(2i(i-2)) = 1/(2(i-2)) = (i+2)/(2(-5)) = **(i+2)/(-10)** At z = -i: Res = -i/((-2i)(-i-2)) = -i/(2i(i+2)) = -1/(2(i+2)) = **(-i+2)/(-10)** At z = 2: Res = 2/((4+1)(1)) = **2/5** Check: Sum = (i+2-i+2)/(-10) + 2/5 = 4/(-10) + 4/10 = 0 ✓ --- ### Problem A4: Essential Singularity Show that e^{1/z} takes every nonzero value infinitely often in any neighborhood of z = 0. **Solution:** For any w ≠ 0, solve e^{1/z} = w: - 1/z = log w + 2πin for n ∈ ℤ - z = 1/(log w + 2πin) For large |n|, these z values become arbitrarily small (close to 0). There are infinitely many solutions for each w ≠ 0. ✓ The exception w = 0 is never achieved since e^{1/z} ≠ 0 for all z ≠ 0. --- ## 🔬 Problem Set B: Residue Theorem & Applications ### Problem B1: Contour Integrals Evaluate: a) ∮_{|z|=3} dz/(z²-1) b) ∮_{|z|=2} z·e^z/(z-1)² dz c) ∮_{|z|=1} e^{1/z} dz **Solutions:** a) Poles at z = ±1, both inside |z| = 3. Res at z=1: 1/(2·1) = 1/2 Res at z=-1: 1/(2·(-1)) = -1/2 ∮ = 2πi(1/2 - 1/2) = **0** b) Double pole at z = 1 inside |z| = 2. Res = d/dz[z·e^z]|_{z=1} = (e^z + z·e^z)|_{z=1} = 2e ∮ = 2πi(2e) = **4πie** c) Essential singularity at z = 0. e^{1/z} = 1 + 1/z + 1/(2z²) + ... Res = coefficient of 1/z = 1 ∮ = 2πi(1) = **2πi** --- ### Problem B2: Real Integrals Evaluate: a) ∫_{-∞}^{∞} dx/(x²+4)² b) ∫_0^{2π} dθ/(2+cos θ) c) ∫_{-∞}^{∞} x·sin x/(x²+1) dx **Solutions:** a) f(z) = 1/(z²+4)² has double poles at z = ±2i. Only z = 2i in upper half-plane. Res = d/dz[(z-2i)²/(z²+4)²]|_{z=2i} = d/dz[1/(z+2i)²]|_{z=2i} = -2/(4i)³ = -2/(-64i) = 1/(32i) ∫ = 2πi · 1/(32i) = **π/16** b) z = e^{iθ}, cos θ = (z+1/z)/2 1/(2+cos θ) = 2z/(z² + 4z + 1) Poles: z = -2 ± √3, only z = -2+√3 inside |z|=1 Res = 2(-2+√3)/(2(-2+√3)+4) = (−2+√3)/(√3) = 1 - 2/√3 ∫ = (2/i)·2πi·Res = **2π/√3** c) Consider ∫ z·e^{iz}/(z²+1) dz. Pole at z = i in upper half-plane. Res = i·e^{-1}/(2i) = e^{-1}/2 ∫_{-∞}^{∞} x·e^{ix}/(x²+1) dx = 2πi · e^{-1}/2 = πi/e Taking imaginary part: **π/e** --- ### Problem B3: Argument Principle a) How many zeros does z⁵ + 3z + 1 have in |z| < 1? b) Prove z⁴ - 5z + 1 has exactly 3 zeros in 1 < |z| < 2. **Solutions:** a) Use Rouché: f(z) = 3z, g(z) = z⁵ + 1 On |z| = 1: |f| = 3, |g| ≤ 2 < 3 f has 1 zero in |z| < 1, so z⁵ + 3z + 1 has **1 zero**. b) Count in |z| < 1: f = -5z, g = z⁴ + 1, |f| = 5, |g| ≤ 2 → **1 zero** Count in |z| < 2: f = z⁴, g = -5z + 1, |f| = 16, |g| ≤ 11 → **4 zeros** Therefore in 1 < |z| < 2: 4 - 1 = **3 zeros** --- ### Problem B4: Physics Application The scattering amplitude is A(E) = 1/(E - E₀ + iΓ/2) with E₀ = 1, Γ = 0.2. a) Where is the pole? b) Compute the cross section σ ∝ |A|² at E = E₀. c) What is the full width at half maximum? **Solutions:** a) Pole at E = E₀ - iΓ/2 = **1 - 0.1i** b) |A(E₀)|² = |1/(iΓ/2)|² = 4/Γ² = 4/0.04 = **100** c) |A(E)|² = 1/((E-E₀)² + Γ²/4) Half max when (E-E₀)² = Γ²/4 E - E₀ = ±Γ/2 FWHM = Γ = **0.2** --- ## 📊 Self-Assessment Rubric Rate yourself 1-5 on each topic: | Topic | Score | Notes | |-------|-------|-------| | Laurent series computation | /5 | | | Singularity classification | /5 | | | Residue calculations | /5 | | | Residue theorem application | /5 | | | Real integral evaluation | /5 | | | Argument principle | /5 | | | Rouché's theorem | /5 | | | Physics connections | /5 | | **Total: /40** - 35-40: Excellent! Ready for Classical Mechanics - 30-34: Good foundation, review weak areas - 25-29: Need more practice - Below 25: Consider additional review --- ## 🎉 Complex Analysis Complete! You've completed **Weeks 19-20: Complex Analysis I & II**! ### What You've Mastered: - Complex numbers and the complex plane - Analyticity and Cauchy-Riemann equations - Elementary complex functions - Contour integration and Cauchy's theorem - Laurent series and singularities - The Residue Theorem - Real integral evaluation - Argument Principle and Rouché's theorem ### Applications in Quantum Mechanics: - Green's functions and propagators - Scattering theory (poles = bound states/resonances) - Kramers-Kronig relations (causality) - Path integrals and analytic continuation - Spectral theory --- ## 🔮 Month 6 Preview: Classical Mechanics **Next:** Weeks 21-24 cover **Classical Mechanics** — the foundation for quantum mechanics! ### Week 21: Lagrangian Mechanics I - Generalized coordinates - Principle of least action - Euler-Lagrange equations - Constraints and Lagrange multipliers ### Week 22: Lagrangian Mechanics II - Symmetries and conservation laws - Noether's theorem - Central force problems - Small oscillations ### Week 23: Hamiltonian Mechanics I - Legendre transformation - Hamilton's equations - Phase space - Poisson brackets ### Week 24: Hamiltonian Mechanics II - Canonical transformations - Hamilton-Jacobi equation - Action-angle variables - Connection to quantum mechanics --- ## ✅ Week 20 Completion Checklist - [ ] Master Laurent series in different annuli - [ ] Classify all singularity types - [ ] Compute residues by multiple methods - [ ] Apply Residue Theorem confidently - [ ] Evaluate real integrals via contours - [ ] Use Argument Principle and Rouché - [ ] Connect to physics applications - [ ] Self-assessment score ≥ 30/40 --- *Congratulations on completing Complex Analysis!* *Ready for the beautiful structure of Classical Mechanics!*

Day 140: Week 20 Review — Complex Analysis II Mastery

Today's Schedule (7 hours)

Day 140: Week 20 Review — Complex Analysis II Mastery

Today's Schedule (7 hours)