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Complex Analysis

42 lessons across 14 topics

1. Complex Numbers and the Complex Plane

1Algebra Of Complex Numbers2Modulus And Argument3Polar Form And Roots

2. Functions of a Complex Variable

4Analyticity5Differentiability6Limits And Continuity

3. Cauchy-Riemann Equations

7Derivation And Interpretation8Examples And Counterexamples9Harmonic Functions

4. Elementary Analytic Functions

10Branch Issues11Exponential And Logarithm12Trigonometric And Hyperbolic Functions

5. Contours and Complex Integration

13Antiderivatives14Estimation15Paths And Contour Integrals

6. Cauchy’s Theorem

16Consequences Of Path Independence17Morera-type Intuition If Included18Simply Connected Domains

7. Cauchy Integral Formula

19Derivatives Of Analytic Functions20Fundamental Theorem Of Algebra21Liouville’s Theorem

8. Midterm 1 and Taylor Series

22Analytic Continuation Intuition23Midterm 124Power Series Representations

9. Laurent Series

25Annuli26Expansion Methods27Principal Parts

10. Singularities

28Essential Singularities29Poles30Removable Singularities

11. Residues

31Applications To Contour Integrals32Residue Computation33Residue Theorem

12. Applications of Residues

34Argument Principle Overview, If Included35Improper Real Integrals36Trigonometric Integrals

13. Conformal Mapping

37Linear Fractional Transformations38Local Angle Preservation39Mapping Regions

14. Final Review

40Key Themes In Final Review41Review And Synthesis42The Structure And Power Of Analytic Functions