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Complex Variables with Applications

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Complex Variables with Applications

Complex analysis is a basic tool in many mathematical theories. By itself and through some of these theories it also has a great many practical applications. There are a small number of far-reaching theorems that we’ll explore in the first part of the class. Along the way, we’ll touch on some mathematical and engineering applications of these theorems. The last third of the class will be devoted to a deeper look at applications.

AnalysisMathematics
Author
Jeremy Orloff
Publisher
Massachusetts Institute of Technology
Source
LibreTexts
License
CC BY-NC-SA
Language
EN
Category
Basic Sciences
Read online Download PDF (open licence)
Contents· 118 sections
  1. 01: Complex Algebra and the Complex Plane
  2. 1.01: Motivation
  3. 1.02: Fundamental Theorem of Algebra
  4. 1.03: Terminology and Basic Arithmetic
  5. 1.04: The Complex Plane
  6. 1.05: Polar Coordinates
  7. 1.06: Euler's Formula
  8. 1.07: The Exponential Function
  9. 1.08: Complex Functions as Mappings
  10. 1.09: The function arg(z)
  11. 1.10: Concise summary of branches and branch cuts
  12. 1.11: The Function log(z)
  13. 1.12: Inverse Euler formula
  14. 1.13: de Moivre's formula
  15. 1.14: Representing Complex Multiplication as Matrix Multiplication
  16. 02: Analytic Functions
  17. 2.01: The Derivative - Preliminaries
  18. 2.02: Open Disks Open Deleted Disks and Open Regions
  19. 2.03: Limits and Continuous Functions
  20. 2.04: The Point at Infinity
  21. 2.05: Derivatives
  22. 2.06: Cauchy-Riemann Equations
  23. 2.07: Cauchy-Riemann all the way down
  24. 2.08: Gallery of Functions
  25. 2.09: Branch Cuts and Function Composition
  26. 2.10: Appendix - Limits
  27. 03: Multivariable Calculus (Review)
  28. 3.01: Terminology and Notation
  29. 3.02: Parametrized curves
  30. 3.03: Chain rule
  31. 3.04: Grad curl and div
  32. 3.05: Level Curves
  33. 3.06: Line Integrals
  34. 3.07: Green's Theorem
  35. 3.08: Extensions and Applications of Greens Theorem
  36. 04: Line Integrals and Cauchys Theorem
  37. 4.01: Introduction to Line Integrals and Cauchys Theorem
  38. 4.02: Complex Line Integrals
  39. 4.03: Fundamental Theorem for Complex Line Integrals
  40. 4.04: Path Independence
  41. 4.05: Examples
  42. 4.06: Cauchy's Theorem
  43. 4.07: Extensions of Cauchy's theorem
  44. 05: Cauchy Integral Formula
  45. 5.01: Cauchy's Integral for Functions
  46. 5.02: Cauchys Integral Formula for Derivatives
  47. 5.03: Proof of Cauchy's integral formula
  48. 5.04: Proof of Cauchy's integral formula for derivatives
  49. 5.05: Amazing consequence of Cauchys integral formula
  50. 06: Harmonic Functions
  51. 6.02: Harmonic Functions
  52. 6.03: Del notation
  53. 6.04: A second Proof that u and v are Harmonic
  54. 6.05: Maximum Principle and Mean Value Property
  55. 6.06: Orthogonality of Curves
  56. 07: Two Dimensional Hydrodynamics and Complex Potentials
  57. 7.01: Velocity Fields
  58. 7.02: Stationary Flows
  59. 7.03: Physical Assumptions and Mathematical Consequences
  60. 7.04: Complex Potentials
  61. 7.05: Stream Functions
  62. 7.06: More Examples with Pretty Pictures
  63. 08: Taylor and Laurent Series
  64. 8.01: Geometric Series
  65. 8.02: Convergence of Power Series
  66. 8.03: Taylor Series
  67. 8.04: Taylor Series Examples
  68. 8.05: Singularities
  69. 8.06: Appendix- Convergence
  70. 8.07: Laurent Series
  71. 8.08: Digression to Differential Equations
  72. 8.09: Poles
  73. 09: Residue Theorem
  74. 9.01: Poles and Zeros
  75. 9.02: Holomorphic and Meromorphic Functions
  76. 9.03: Behavior of functions near zeros and poles
  77. 9.04: Residues
  78. 9.05: Cauchy Residue Theorem
  79. 9.06: Residue at
  80. 10: Definite Integrals Using the Residue Theorem
  81. 10.01: Integrals of functions that decay
  82. 10.02: Integrals
  83. 10.03: Trigonometric Integrals
  84. 10.04: Integrands with branch cuts
  85. 10.05: Cauchy principal value
  86. 10.06: Integrals over portions of circles
  87. 10.07: Fourier transform
  88. 10.08: Solving DEs using the Fourier transform
  89. 11: Conformal Transformations
  90. 11.01: Geometric Definition of Conformal Mappings
  91. 11.02: Tangent vectors as complex numbers
  92. 11.03: Analytic functions are Conformal
  93. 11.04: Digression to harmonic functions
  94. 11.05: Riemann Mapping Theorem
  95. 11.06: Examples of conformal maps and excercises
  96. 11.07: Fractional Linear Transformations
  97. 11.08: Reflection and symmetry
  98. 11.09: Flows around cylinders
  99. 11.10: Solving the Dirichlet problem for harmonic functions
  100. 12: Argument Principle
  101. 12.01: Principle of the Argument
  102. 12.02: Nyquist Criterion for Stability
  103. 12.03: A Bit on Negative Feedback
  104. 13: Laplace Transform
  105. 13.01: A brief introduction to linear time invariant systems
  106. 13.02: Laplace transform
  107. 13.03: Exponential Type
  108. 13.04: Properties of Laplace transform
  109. 13.05: Differential equations
  110. 13.06: Table of Laplace transforms
  111. 13.07: System Functions and the Laplace Transform
  112. 13.08: Laplace inverse
  113. 13.09: Delay and Feedback
  114. 14: Analytic Continuation and the Gamma Function
  115. 14.01: Analytic continuation
  116. 14.02: Definition and properties of the Gamma function
  117. 14.03: Connection to Laplace
  118. 14.04: Proofs of (some) properties