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Nydus Open Learning

Mathematics

Mathematics · Advanced · Level 3

Complex analysis

Complex numbers in the plane, with modulus, conjugate, polar form, de Moivre's theorem and the roots of unity; the exponential and the logarithm with its branches; limits along every direction at once, the complex derivative, the Cauchy-Riemann equations, harmonic functions and harmonic conjugates; contours, contour integrals and the estimation bound, Cauchy's theorem, Cauchy's integral formula and the theorems Liouville's argument buys; Taylor and Laurent series, the classification of isolated singularities, residues and the residue theorem, with the real integrals it settles; and conformal maps — angle preservation, Möbius transformations, the cross ratio, circles and lines, and the mapping of regions that turns a hard boundary value problem into an easy one.

Ages 18+ 24 lessons about 10 hours in all

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The lessons

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  1. 1 Complex arithmetic in the plane
  2. 2 Modulus, conjugate and distance
  3. 3 Polar form and de Moivre's theorem
  4. 4 Roots and roots of unity
  5. 5 The complex exponential and logarithm
  6. 6 Limits and continuity in the plane
  7. 7 The complex derivative
  8. 8 The Cauchy-Riemann equations
  9. 9 Harmonic functions
  10. 10 Harmonic conjugates
  11. 11 Contours and winding numbers
  12. 12 Contour integrals and the estimation bound
  13. 13 Cauchy's theorem
  14. 14 Cauchy's integral formula
  15. 15 Liouville's theorem and its consequences
  16. 16 Taylor series of an analytic function
  17. 17 Laurent series
  18. 18 Classifying isolated singularities
  19. 19 Residues
  20. 20 The residue theorem and real integrals
  21. 21 Conformal maps
  22. 22 Möbius transformations
  23. 23 The cross ratio, circles and lines
  24. 24 Mapping regions and the Dirichlet problem