Math 2.0

Learn Year 2

Complex analysis

84 ideas in teaching order. Each idea comes after the ideas that it builds on.

Start with Domain in the complex plane →

  1. Domain in the complex plane definition
  2. Complex differentiability definition
  3. Holomorphic function definition
  4. Entire function definition
  5. Cauchy–Riemann equations statement
  6. The real part of a holomorphic function is harmonic statement
  7. Harmonic conjugate definition
  8. The complex conjugate is not holomorphic counterexample
  9. Complex exponential definition
  10. The complex exponential is entire statement
  11. Derivative of the complex exponential statement
  12. The complex exponential is 2πi-periodic statement
  13. exp z = 1 iff z is an integer multiple of 2πi statement
  14. Complex sine and cosine definition
  15. Power series of complex cosine statement
  16. Multivalued function concept
  17. Complex logarithm definition
  18. Branch of the logarithm definition
  19. Principal branch definition
  20. Branch cut definition
  21. The principal logarithm is holomorphic off the branch cut statement
  22. Complex power definition
  23. Power series in the complex plane definition
  24. Disc of convergence definition
  25. Power series are holomorphic in the disc of convergence statement
  26. Power series of the complex logarithm statement
  27. Contour integral definition
  28. Circle integral definition
  29. Estimation lemma statement
  30. Fundamental theorem of contour integration statement
  31. Integral of 1/z around the unit circle example
  32. Goursat's theorem statement
  33. Cauchy's integral theorem statement
  34. Holomorphic functions on a disc have primitives statement
  35. Cauchy's integral formula statement
  36. Holomorphic functions are analytic statement
  37. Holomorphic functions are infinitely differentiable statement
  38. Cauchy's integral formula for derivatives statement
  39. Cauchy's estimates statement
  40. Liouville's theorem statement
  41. Fundamental theorem of algebra statement
  42. Morera's theorem statement
  43. Mean value property statement
  44. Uniform limits of holomorphic functions are holomorphic statement
  45. Zero of a holomorphic function definition
  46. Order of a zero definition
  47. Isolated zeros theorem statement
  48. Identity theorem statement
  49. Analytic continuation concept
  50. Maximum modulus principle statement
  51. Open mapping theorem statement
  52. Schwarz lemma statement
  53. Annulus definition
  54. Laurent series definition
  55. Existence of Laurent expansions on an annulus statement
  56. Principal part definition
  57. Isolated singularity definition
  58. Removable singularity definition
  59. Riemann's theorem on removable singularities statement
  60. Pole definition
  61. Order of a pole definition
  62. Essential singularity definition
  63. exp(1/z) has an essential singularity at 0 example
  64. Casorati–Weierstrass theorem statement
  65. Picard's great theorem statement
  66. Meromorphic function definition
  67. Residue definition
  68. Residue at a simple pole statement
  69. Winding number definition
  70. Residue theorem statement
  71. Evaluation of real integrals by residues concept
  72. Jordan's lemma statement
  73. Cauchy principal value definition
  74. Argument principle statement
  75. Rouché's theorem statement
  76. Hurwitz's theorem statement
  77. Conformal map definition
  78. Holomorphic functions with nonzero derivative are conformal statement
  79. Riemann sphere definition
  80. Möbius transformation definition
  81. Möbius transformations map circles and lines to circles and lines statement
  82. Cross-ratio definition
  83. Cayley transform example
  84. Riemann mapping theorem statement