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Complex Methods for the Sciences

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Complex Methods for the Sciences

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).

Mathematical Physics and PedagogyPhysics
Author
Y. D. Chong
Publisher
Nanyang Technological University
Source
LibreTexts
License
CC BY-SA
Language
EN
Category
Basic Sciences
Read online Download PDF (open licence)
Contents· 78 sections
  1. 01: Mathematical Functions
  2. 1.01: Real Functions
  3. 1.02: 1.2 The Exponential Function
  4. 1.03: 1.3 The Logarithm Function
  5. 1.04: Non-Natural Powers
  6. 1.05: 1.5 Trigonometric Functions
  7. 1.06: Hyperbolic Functions
  8. 1.07: Continuity
  9. 1.08: Exercises
  10. 02: Derivatives
  11. 2.01: Properties of Derivatives
  12. 2.02: Taylor Series
  13. 2.03: Ordinary Differential Equations
  14. 2.04: Partial Derivatives
  15. 2.05: Exercises
  16. 03: Integrals
  17. 3.01: Basic Properties of Definite Integrals
  18. 3.02: Integrals as Antiderivatives
  19. 3.03: Integration by Parts
  20. 3.04: Change of Variables
  21. 3.05: The Gaussian Integral
  22. 3.06: Differentiating Under the Integral Sign
  23. 3.07: Exercises
  24. 04: Complex Numbers
  25. 4.01: Complex Algebra
  26. 4.02: Conjugates and Magnitudes
  27. 4.03: Euler's Formula
  28. 4.04: The Complex Plane
  29. 4.05: Complex Functions
  30. 4.06: Trajectories in the Complex Plane
  31. 4.07: Why Complex Numbers
  32. 4.08: Exercises
  33. 05: Complex Oscillations
  34. 5.01: The Damped Harmonic Oscillator
  35. 5.02: Complex Solution
  36. 5.03: General Solution for the Damped Harmonic Oscillator
  37. 5.04: Stating the Solution in Terms of Initial Conditions
  38. 5.05: Exercises
  39. 06: Complex Waves
  40. 6.01: The Wave Equation
  41. 6.02: Real Solutions to the Wave Equation
  42. 6.03: Complex Solutions to the Wave Equation
  43. 6.04: Waves in 3D Space
  44. 6.05: Harmonic Waves
  45. 6.06: Exercises
  46. 07: Complex Derivatives
  47. 7.01: Complex Continuity and Differentiability
  48. 7.02: Analytic Functions
  49. 7.03: The Cauchy-Riemann Equations
  50. 7.04: Exercises
  51. 08: Branch Points and Branch Cuts
  52. 8.01: Non-Integer Powers as Multi-Valued Operations
  53. 8.02: Branches
  54. 8.03: Aside- The Meaning of Infinity for Complex Numbers
  55. 8.04: Branch Cuts for General Multi-Valued Operations
  56. 8.05: Exercises
  57. 09: Contour Integration
  58. 9.01: Contour Integrals
  59. 9.02: Cauchy's Integral Theorem
  60. 9.03: Poles
  61. 9.04: Using Contour Integration to Solve Definite Integrals
  62. 9.05: Exercises
  63. 10: Fourier Series and Fourier Transforms
  64. 10.01: Fourier Series
  65. 10.02: Fourier Transforms
  66. 10.03: Fourier Transforms for Time-Domain Functions
  67. 10.04: Basic Properties of the Fourier Transform
  68. 10.05: Fourier Transforms of Differential Equations
  69. 10.06: Common Fourier Transforms
  70. 10.07: The Delta Function
  71. 10.08: Multi-Dimensional Fourier Transforms
  72. 10.09: Exercises
  73. 11: Green's Functions
  74. 11.01: The Driven Harmonic Oscillator
  75. 11.02: Space-Time Green's Functions
  76. 11.03: Causality and the Time-Domain Green's Function
  77. 11.04: Looking Ahead
  78. 11.05: Exercises