Complex Methods for the Sciences

by Y. D. Chong

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An introductory course in mathematical methods for science and engineering students. Topics covered include complex analysis, contour integration, Fourier analysis, and Green's function methods. Mathematical methods are illustrated with examples drawn from physics, with a focus on oscillations and waves. These notes are also available in the form of Jupyter notebooks on Github (GPLv3).

Description from LibreTexts.

Publisher
Nanyang Technological University
Subjects
Physics, Mathematical Physics and Pedagogy
Licence
CC BY-SA 4.0, as LibreTexts states it
Who can read
Tweens profiles and older, judged from its title and subjects

Contents

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  1. Front Matter
  2. TitlePage
  3. InfoPage
  4. Table of Contents
  5. Licensing
  6. 1: Mathematical Functions
  7. 1.1: Real Functions
  8. 1.2: 1.2 The Exponential Function
  9. 1.3: 1.3 The Logarithm Function
  10. 1.4: Non-Natural Powers
  11. 1.5: 1.5 Trigonometric Functions
  12. 1.6: Hyperbolic Functions
  13. 1.7: Continuity
  14. 1.8: Exercises
  15. 2: Derivatives
  16. 2.1: Properties of Derivatives
  17. 2.2: Taylor Series
  18. 2.3: Ordinary Differential Equations
  19. 2.4: Partial Derivatives
  20. 2.5: Exercises
  21. 3: Integrals
  22. 3.1: Basic Properties of Definite Integrals
  23. 3.2: Integrals as Antiderivatives
  24. 3.3: Integration by Parts
  25. 3.4: Change of Variables
  26. 3.5: The Gaussian Integral
  27. 3.6: Differentiating Under the Integral Sign
  28. 3.7: Exercises
  29. 4: Complex Numbers
  30. 4.1: Complex Algebra
  31. 4.2: Conjugates and Magnitudes
  32. 4.3: Euler's Formula
  33. 4.4: The Complex Plane
  34. 4.5: Complex Functions
  35. 4.6: Trajectories in the Complex Plane
  36. 4.7: Why Complex Numbers?
  37. 4.8: Exercises
  38. 5: Complex Oscillations
  39. 5.1: The Damped Harmonic Oscillator
  40. 5.2: Complex Solution
  41. 5.3: General Solution for the Damped Harmonic Oscillator
  42. 5.4: Stating the Solution in Terms of Initial Conditions
  43. 5.5: Exercises
  44. 6: Complex Waves
  45. 6.1: The Wave Equation
  46. 6.2: Real Solutions to the Wave Equation
  47. 6.3: Complex Solutions to the Wave Equation
  48. 6.4: Waves in 3D Space
  49. 6.5: Harmonic Waves
  50. 6.6: Exercises
  51. 7: Complex Derivatives
  52. 7.1: Complex Continuity and Differentiability
  53. 7.2: Analytic Functions
  54. 7.3: The Cauchy-Riemann Equations
  55. 7.4: Exercises
  56. 8: Branch Points and Branch Cuts
  57. 8.1: Non-Integer Powers as Multi-Valued Operations
  58. 8.2: Branches
  59. 8.3: Aside- The Meaning of "Infinity" for Complex Numbers
  60. 8.4: Branch Cuts for General Multi-Valued Operations
  61. 8.5: Exercises
  62. 9: Contour Integration
  63. 9.1: Contour Integrals
  64. 9.2: Cauchy's Integral Theorem
  65. 9.3: Poles
  66. 9.4: Using Contour Integration to Solve Definite Integrals
  67. 9.5: Exercises
  68. 10: Fourier Series and Fourier Transforms
  69. 10.1: Fourier Series
  70. 10.2: Fourier Transforms
  71. 10.3: Fourier Transforms for Time-Domain Functions
  72. 10.4: Basic Properties of the Fourier Transform
  73. 10.5: Fourier Transforms of Differential Equations
  74. 10.6: Common Fourier Transforms
  75. 10.7: The Delta Function
  76. 10.8: Multi-Dimensional Fourier Transforms
  77. 10.9: Exercises
  78. 11: Green's Functions
  79. 11.1: The Driven Harmonic Oscillator
  80. 11.2: Space-Time Green's Functions
  81. 11.3: Causality and the Time-Domain Green's Function
  82. 11.4: Looking Ahead
  83. 11.5: Exercises
  84. Back Matter
  85. Index
  86. Glossary
  87. Detailed Licensing