Part I: The Complex Plane and Elementary Functions
1. Complex Numbers & Properties
- Algebra of Complex Numbers ($z = x + iy$) [Click Here]
- Modulus and Conjugate [Click Here]
- Polar Form & Argument ($re^{i\theta}$) [Click Here]
- De Moivre’s Theorem & Roots of Unity [Click Here]
- Topology of Complex Plane (Open/Closed sets) [Click Here]
- geometry : stereographic projection [Click Here]
2. Functions of a Complex Variable
- Functions of complex variable [Click Here]
- Limits of Functions [Click Here]
- Continuity [Click Here]
- Differentiability [Click Here]
- Cauchy-Riemann (C-R) Equations (Cartesian & Polar) [Click Here]
- Analytic (Holomorphic) Functions [Click Here]
- properties of analytic function [Click Here]
- Harmonic Functions & Harmonic Conjugates [Click Here]
- Construction of Analytic Function (Milne-Thomson Method) [Click Here]
3. Elementary Functions
- Exponential Function ($e^z$) [Click Here]
- Logarithmic Function & Branches ($\log z$) [Click Here]
- Trigonometric Functions ($\sin z, \cos z$) [Click Here]
- Hyperbolic Functions ($\sinh z, \cosh z$) [Click Here]
- Inverse Trigonometric & Hyperbolic Functions [Click Here]
MULTI BRANCH FUNCTION