Methods of Applied Mathematics Lecture Notes

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$Math 520a - Midterm take home exam Do 5 out of the 6 problems ...$

$page 1 of 2 Math 410 Matrix Analysis Fall 2007 Final Exam Answers ...$

4 CONTENTS2 Fourier series 372.1 Orthonormal families . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 L 2 convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.3 Absolute convergence . . . . . . . . . . . . . . . . . . . . . . . . . 402.4 Pointwise convergence . . . . . . . . . . . . . . . . . . . . . . . . 412.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Fourier transforms 453.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 L 1 theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3 L 2 theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4 Absolute convergence . . . . . . . . . . . . . . . . . . . . . . . . . 473.5 Fourier transform pairs . . . . . . . . . . . . . . . . . . . . . . . . 483.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.7 Poisson summation formula . . . . . . . . . . . . . . . . . . . . . 503.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.9 PDE Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 Complex integration 534.1 Complex number quiz . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Complex functions . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2.1 Closed and exact forms . . . . . . . . . . . . . . . . . . . 544.2.2 Cauchy-Riemann equations . . . . . . . . . . . . . . . . . 554.2.3 The Cauchy integral theorem . . . . . . . . . . . . . . . . 554.2.4 Polar representation . . . . . . . . . . . . . . . . . . . . . 564.2.5 Branch cuts . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3 Complex integration and residue calculus . . . . . . . . . . . . . 574.3.1 The Cauchy integral formula . . . . . . . . . . . . . . . . 574.3.2 The residue calculus . . . . . . . . . . . . . . . . . . . . . 584.3.3 Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3.4 A residue calculation . . . . . . . . . . . . . . . . . . . . . 594.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.5 More residue calculus . . . . . . . . . . . . . . . . . . . . . . . . 604.5.1 Jordan’s lemma . . . . . . . . . . . . . . . . . . . . . . . . 604.5.2 A more delicate residue calculation . . . . . . . . . . . . . 614.5.3 Cauchy formula for derivatives . . . . . . . . . . . . . . . 614.5.4 Poles of higher order . . . . . . . . . . . . . . . . . . . . . 624.5.5 A residue calculation with a double pole . . . . . . . . . . 624.6 The Taylor expansion . . . . . . . . . . . . . . . . . . . . . . . . 634.6.1 Radius of convergence . . . . . . . . . . . . . . . . . . . . 634.6.2 Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . 64
CONTENTS 55 Distributions 675.1 Properties of distributions . . . . . . . . . . . . . . . . . . . . . . 675.2 Mapping distributions . . . . . . . . . . . . . . . . . . . . . . . . 695.3 Radon measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.4 Approximate delta functions . . . . . . . . . . . . . . . . . . . . . 705.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.6 Tempered distributions . . . . . . . . . . . . . . . . . . . . . . . . 725.7 Poisson equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.8 Diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . 755.9 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.10 Homogeneous solutions of the wave equation . . . . . . . . . . . 775.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.12 Answers to first two problems . . . . . . . . . . . . . . . . . . . . 796 Bounded Operators 816.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2 Bounded linear operators . . . . . . . . . . . . . . . . . . . . . . 816.3 Compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . 846.4 Hilbert-Schmidt operators . . . . . . . . . . . . . . . . . . . . . . 876.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.6 Finite rank operators . . . . . . . . . . . . . . . . . . . . . . . . . 896.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907 Densely Defined Closed Operators 937.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937.2 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937.3 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947.4 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957.5 The spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957.6 Spectra of inverse operators . . . . . . . . . . . . . . . . . . . . . 967.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977.8 Self-adjoint operators . . . . . . . . . . . . . . . . . . . . . . . . . 987.9 First order differential operators with a bounded interval: pointspectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987.10 Spectral projection and reduced resolvent . . . . . . . . . . . . . 1007.11 Generating second-order self-adjoint operators . . . . . . . . . . . 1017.12 First order differential operators with a semi-infinite interval:residual spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.13 First order differential operators with an infinite interval: continuousspectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.15 A pathological example . . . . . . . . . . . . . . . . . . . . . . . 104
Page 1: Methods of Applied MathematicsLectu
Page 6 and 7: 6 CONTENTS8 Normal operators 1058.1
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Page 20 and 21: 20 CHAPTER 1. LINEAR ALGEBRAWe thin
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54 CHAPTER 4. COMPLEX INTEGRATION4.
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56 CHAPTER 4. COMPLEX INTEGRATIONEx
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64 CHAPTER 4. COMPLEX INTEGRATIONBu
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66 CHAPTER 4. COMPLEX INTEGRATION
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68 CHAPTER 5. DISTRIBUTIONSin that
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70 CHAPTER 5. DISTRIBUTIONS5.3 Rado
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72 CHAPTER 5. DISTRIBUTIONS7. Take
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74 CHAPTER 5. DISTRIBUTIONS5.7 Pois
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76 CHAPTER 5. DISTRIBUTIONSIt says
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78 CHAPTER 5. DISTRIBUTIONSand its
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80 CHAPTER 5. DISTRIBUTIONSSince th
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82 CHAPTER 6. BOUNDED OPERATORSIt i
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84 CHAPTER 6. BOUNDED OPERATORSExam
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6.4. HILBERT-SCHMIDT OPERATORS 87Th
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6.6. FINITE RANK OPERATORS 895. Let
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6.7. PROBLEMS 914. Consider functio
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Chapter 7Densely Defined ClosedOper
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7.4. OPERATORS 95Corollary. R(L) =
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7.7. PROBLEMS 977.7 ProblemsIf K is
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7.9. FIRST ORDER DIFFERENTIAL OPERA
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7.11. GENERATING SECOND-ORDER SELF-
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7.14. PROBLEMS 103Example: Let H be
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Chapter 8Normal operators8.1 Spectr
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8.3. VARIATION OF PARAMETERS AND GR
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8.5. SECOND ORDER DIFFERENTIAL OPER
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8.7. THE SPECTRAL THEOREM FOR NORMA
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8.10. EXAMPLES: SCHRÖDINGER OPERAT
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8.11. SUBNORMAL OPERATORS 115plane.
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8.12. EXAMPLES: FORWARD TRANSLATION
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8.14. PROBLEMS 119If we assume that
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Chapter 9Calculus of Variations9.1
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9.3. SECOND VARIATION 123Example 1:
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9.4. INTERLUDE: THE LEGENDRE TRANSF
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9.6. HAMILTONIAN MECHANICS 127Defin
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9.8. PROBLEMS 1299.8 ProblemsLet L
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9.10. APPENDIX: LAGRANGE MULTIPLIER
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Chapter 10Perturbation theory10.1 T
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10.2. PROBLEMS 135So the expansion
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10.4. NONLINEAR DIFFERENTIAL EQUATI
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10.6. EIGENVALUES AND EIGENVECTORS
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10.7. THE SELF-ADJOINT CASE 141Henc
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10.8. THE ANHARMONIC OSCILLATOR 143
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