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source: James Cook 2015年8月24日

Complex Analysis of 2015

This is the playlist for lectures captured from Math 331 of Liberty University of Lynchburg VA during the Fall 2015 semester. This course follows part of Gamelin's Complex Analysis. This introduction to complex analysis is intended for someone who has completed the calculus sequence and has some experience with proof. However, there is no real analysis prerequisite so we do not chase all the rabbits... just some. Anyway, see: http://www.supermath.info/Complex.html.

In a nutshell, we study complex number models, basic topology and limit theory in the complex plane, nth roots of unity, the complex exponential, log, sine, cosine, cosh, sinh, the CR equations and the relation of complex-linearity of the differential, holomorphic functions, conformal functions, basic idea of analytic continuation, Riemann Sphere, harmonic functions and conjugates, Green's Theorem complexified, Cauchy integral theorem and formulas, Goursat Theorem, Lioville Theorem, basic theory of complex power series, Laurent series, residue theory, techniques of complex integration, conformal mapping and fluid flow (perhaps from Fisher's text which is available as a Dover edition) , Mandelbrot and Julia set, and select topics TBD in part by interest of students.

Complex Analysis: Lecture 1: complex numbers 28:39 Here we discuss a few possible models for the complex numbers. Our choice, for convenience, is that the complex numbers are the plane with the Gauss multiplication. That said, we use the notation a+ib in almost everything which follows. Complex conjugation is defined and connections to the real and imaginary part maps are given. We also discuss why every nonzero complex number has a multiplicative inverse (but, I stop short of mentioning that makes C a field, more on that next time). We are about half-way through section 1.1 of http://www.supermath.info/GuideToGame... also the course page is at http://www.supermath.info/Complex.html

Lecture 2: imaginary exponential and nth roots 54:07

Lecture 3: branch cuts, complex exponential 55:02

Lecture 4: more on square root and trig and hyperbolics 54:01

Lecture 5: sequential limits, open sets 45:38

Lecture 6: topological terms and complex differentiability 51:06

Lecture 7: Caratheodory applied and CR equations 52:35

Lecture 8: partial proof of Cauchy Riemann result 55:28

Lecture 9: CR equations and Inverse Function Theorem 51:13

Lecture 10: harmonic functions 48:54

Lecture 11: conformality and FLTs 49:55

Lecture 12: circles and lines and infinity fun with FLT 49:15

Lecture 13: solution to quiz 1 50:20

Lecture 14: exact forms and real integration 50:03

Lecture 15: closed and exact forms 53:41

Lecture 16: closed forms and harmonic conjugates 50:27

Lecture 17 : fluid flow, circulation and flux, Laplace equation 54:01

Lecture 18: Laplace Solns, Complex Integral Introduced 53:30

Lecture 19: basics of complex integration 53:46

Lecture 20: Cauchy's Theorem and Integral Formula 56:19

Lecture 21: examples of applying Cauchy's Integral Formula 50:02

Lecture 22: an example with partial fractions, Morera and Goursat proofs 53:26

Lecture 23: solution to Quiz 2 51:41

Lecture 24: review for Test 2 48:58

Lecture 25: sequences, convergence 48:29

Lecture 26: domain of power series 50:26

Lecture 27: power series calculation 49:36

Lecture 28: power series calculation, order of zeros 55:08

Lecture 29: analytic continuation elementary comments 51:12

Lecture 29.5: singularity theorems 38:57

Lecture 30: partial fractions, residues 53:42

Lecture 31: Cauchy's Residue Thm and the 4 rules 51:03

Lecture 32: integration via residue theory 49:05

Lecture 33: argument principal, Rouche's Theorem 51:35

Lecture 34 part 1: comment on non-Euclidean geometry, Mittag Leffler exhibited 11:29

Lecture 34 part 2: some infinite product calculations 37:12

Lecture 35: series calculation via residues 44:33

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