Learn Complex Analysis
On your own!
The math behind it:
- The limit definition of a derivative
- Mathematical definitions of continuity and convergence
- Convergence tests
- Integration, Fundamental Theorem of Calculus, integration tricks and multi variable integration
- Green’s Theorem and identities
- Partial derivatives
- Harmonic functions and why they are cool
- Complex numbers and arithmetic
- Euiler’s identity
- Taylor series and radii of convergence
- Some R^2 related set theory terms like “open”, “neighborhood”, “domain”, and “bounded”
- Curve related terms like Jordan, simple, closed, and k-connected
With those, you could then understand actual complex analysis:
- What analytic functions are and why we care
- Notation like f, F, z, x, y, omega, gamma, capital gamma, alpha, theta, r, epsilon
- Ways to think about analytic functions that are useful, like splitting them up, in terms of polar co-ordinates, etc.
- The Cauchy-Riemann Equations (duh)
- The link between Harmonicity and Analyticity, harmonic conjugates.
- Sturm and louiville theorems
- Maximum modulus principle, minimum modulus, strong/weak variations
- The ML inequality
- Integration in complex numbers
- Playing with du, dz, d$\alpha$ and dF
- Cauchy Integral theorem
- Cauchy Integral formula (Amazing!)
- Cauchy Integral formula for higher derivatives of f (EVEN BETTER!)
- Analyticity implies infinite derivatives
- Analyticity implies many nice things
- Cauchy convergence criterion
- Uniform convergence (UC)
- Weirstrass M test
- If a sequence of continuous, differentiable, or analytic functions UC, then it is to a continuous, differentiable, or analytic function.
- Absolute convergence means you can fiddle with the order of the terms and other things
- Analytic functions yeild power series
- Power series yeild Analytic functions
- Analyticity and Taylor series
- Removable discontinuities/poles
- Residue theorem!!!!
I also highly suggest the cheap, highly readable textbook: “Complex Variables: Harmonic and Analytic Functions” by Francis J Flanigan.