[CA/Week 5] 8. Riemann surfaces of the less trivial algebraic functions.

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Week 5 of the course "Complex Analysis" is dedicated to analytic continuation and Riemann surfaces

In this video we discuss the Riemann surface of sqrt{1-z^2} and compute the integral along the contour spanning two Riemann sheets.
Topics of the course:

1. Algebra of complex numbers. Differentiation and integration in a complex plane.
2. Singularities of functions of complex variables.
3. Contour integration. Residue theory. Jordan's lemma.
4. Multivalued functions and regular branches
5. Analytical continuation and Riemann surfaces.
6. Integrals with multivalued functions.
7. Gamma function and its applications.
  1. Stokes-Line
  2. Analytic continuation
  3. Analytical continuation
  4. Contour deformation
  5. Complex analysis
  6. Contour integral
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Nice graphics, thank you for that.
What I'm always wondering is: why not 'simply' show the function surface itself in its 4D space? (the different branches come 'naturally' with the different function values; no need to 'design' a Riemann abscis plane).
Surfaces in 4D space can be graphed the same way as surfaces in 3D space! (obviously, some information is lost through the projection procedures, but what you get is the function surfaces themselves, not some 3D extraction like Re or Im, or just an abscis and ordinate plane map).
19: w=z^a
21: w=1/z^a
22: w=ln z
28: w=z^z
32: Lambert W function
34: 'Circle-Hyperbola' functions

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