# Learn Complex Analysis

The math behind it:

1. The limit definition of a derivative
2. Mathematical definitions of continuity and convergence
3. Convergence tests
4. Integration, Fundamental Theorem of Calculus, integration tricks and multi variable integration
5. Green’s Theorem and identities
6. Partial derivatives
7. Harmonic functions and why they are cool
8. Complex numbers and arithmetic
9. Euiler’s identity
10. Taylor series and radii of convergence
11. Some R^2 related set theory terms like “open”, “neighborhood”, “domain”, and “bounded”
12. Curve related terms like Jordan, simple, closed, and k-connected

With those, you could then understand actual complex analysis:

1. What analytic functions are and why we care
2. Notation like f, F, z, x, y, omega, gamma, capital gamma, alpha, theta, r, epsilon
3. Ways to think about analytic functions that are useful, like splitting them up, in terms of polar co-ordinates, etc.
4. The Cauchy-Riemann Equations (duh)
5. The link between Harmonicity and Analyticity, harmonic conjugates.
6. Sturm and louiville theorems
7. Maximum modulus principle, minimum modulus, strong/weak variations
8. The ML inequality
9. Integration in complex numbers
10. Playing with du, dz, d$\alpha$ and dF
11. Cauchy Integral theorem
12. Cauchy Integral formula (Amazing!)
13. Cauchy Integral formula for higher derivatives of f (EVEN BETTER!)
14. Analyticity implies infinite derivatives
15. Analyticity implies many nice things
16. Cauchy convergence criterion
17. Uniform convergence (UC)
18. Weirstrass M test
19. If a sequence of continuous, differentiable, or analytic functions UC, then it is to a continuous, differentiable, or analytic function.
20. Absolute convergence means you can fiddle with the order of the terms and other things
21. Analytic functions yeild power series
22. Power series yeild Analytic functions
23. Analyticity and Taylor series
24. Removable discontinuities/poles
25. Residue theorem!!!!
26. ???
27. Profit

I also highly suggest the cheap, highly readable textbook: “Complex Variables: Harmonic and Analytic Functions” by Francis J Flanigan.