Intro Complex Analysis Lec 21, Conformality, Riemann Mapping Theorem, Vector Fields, Integration

Pictures of the class were being taken the first few minutes. Also sorry about the clicking noises from the camera tripod.

Lecture 21. (0:00) Upcoming schedule and project information. (1:11) Amplitwist and Chain-Rule related reasoning for why analytic mappings are conformal when the derivative is nonzero. (7:10) Riemann mapping theorem statement. (10:43) Prelude to complex integration: complex-valued functions as vector fields and their graphs. (17:03) The vector field "rotates" around circles, and this can be used to define the index of a vector field at a singularity (which can be taken to be a zero or a pole). (25:36) It will be more natural to view complex functions as vector fields for the purposes of integration and the applications will switch over to work done by a force, flux of a (2-dimensional) fluid flow across a curve (1-dimensional) membrane. (26:56) The squaring function as a vector field. (28:41) Integrating a complex function over a (parabolic) contour via a parameterization. (37:43) Doing the same integral with the complex version of the Fundamental Theorem of Calculus. (42:23) Approximating the answer via a Riemann sum and a start at visualizing it.

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[adriannowak]

Im about to code a tool that lets you map custom drawn simply connected domains onto the open unit disk. What is the best way to compute this? Ive already studied the neumann kernel where youll get an integral equation and a constructive proof given by caratheodory and koebe where you iterate through domains but that converges kinda slowly. Any hints?

[ahmedgharib]

what amazing
complex analysis became so easy
thanks so much professor