Complex Analysis

Advanced Mathematics by Maths 505. 36 lectures, 8.3 hours. Covers: complex numbers analytic functions contour integrals residues cauchy.

Lectures

1. An unorthodox intro to complex numbers · 14 min
2. A nice little proof for Euler's formula · 8 min
3. complex analysis: polar forms and how branch cuts work · 14 min
4. Complex Analysis: complex valued functions and some important concepts · 10 min
5. limits and continuity in complex analysis · 8 min
6. Complex Analysis: Powers and Nth roots · 19 min
7. the complex exponential function · 10 min
8. The fascinating case of the complex logarithm · 18 min
9. Complex Analysis: Power Functions · 25 min
10. Complex Analysis: trigonometric and inverse trigonometric functions · 15 min
11. complex differentiation · 10 min
12. Complex Analysis: what is an analytic function? · 25 min
13. Complex Analysis: harmonic functions and differential operators · 8 min
14. Complex Analysis: Conformal Mappings · 8 min
15. Complex Analysis: what is a contour integral? · 10 min
16. Complex Analysis: the fundamental theorem and independence of path for contour integrals · 17 min
17. Complex Analysis: Gauss' Mean Value Theorem · 9 min
18. What are the residues of the gamma function? · 7 min
19. A surprisingly complex trigonometric integral: int (0,π/2) 1/(1+sin^4(x)) · 12 min
20. 2 gorgeous results: derivatives of the eta and zeta functions at zero · 12 min
21. int 0 to infinity (cos(x))/(π^2 - 4x^2): the best thing you'll see next week · 16 min
22. This integral breaks math · 7 min
23. Why this integral is MUCH harder than it looks · 21 min
24. The fresnel integrals solved using contour integration · 14 min
25. A surprisingly interesting integral · 10 min
26. A (literally) complex integral · 12 min
27. one of the best integrals you'll solve · 14 min
28. This wacky integral has a beautiful result · 12 min
29. Contour integration with a STUNNING result! · 16 min
30. This is one of the coolest integrals ever solved · 9 min
31. A simple looking integral with a complex solution · 12 min
32. A wild complex integral! · 12 min
33. A stellar integral solved using some wonderful complex analysis · 20 min
34. My favorite contour integration problem! · 29 min
35. This result is FANTASTIC · 15 min
36. Complex Analysis: What is the residue theorem? · 14 min

Quick explainers

• The 5 ways to visualize complex functions | Essence of complex analysis #3 · 15 min
• What does a complex function look like? #SoME3 · 21 min
• But what is the Riemann zeta function? Visualizing analytic continuation · 22 min
• Visualizing Complex-Valued Functions · 24 min
• What does it mean to take a complex derivative? (visually explained) · 25 min
• Visualizing Complex Integrals · 9 min
• The shocking connection between complex numbers and geometry. · 14 min
• e^(iπ) in 3.14 minutes, using dynamics | DE5 · 4 min
• Imaginary Numbers Are Real [Part 1: Introduction] · 6 min
• Imaginary Numbers, Functions of Complex Variables: 3D animations. · 15 min

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