   Exponential & Reciprocal Maps, Domains, Derivative Limit Calculations by
Video Lecture 7 of 35
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Date Added: July 29, 2017 Lights Out New Window

### Lecture Description

(0:00) Quiz 2 due date and topics. (0:30) Mathematica demonstration of the stereographic projection onto the Riemann sphere. (4:09) Review the formula for the exponential mapping and animate images of rectangles on Mathematica. (12:28) e^(z) is periodic with period 2*pi*i. (13:51) Making sense of the image from the formula. (15:04) Reciprocal mapping and algebraically verifying images of lines and circles. (28:51) Definition of a "domain" in complex analysis as an open connected subset of the complex plane (a typical "domain of definition" for functions in complex analysis). (30:21) What does it mean for a set to be (path) connected? (32:33) What does it mean for a set to be open (in terms of interior points)? Demonstrate idea with Mathematica. (39:52) Derivative definition and limit calculations for f(z) = z^2 and f(z) = 1/z.

### Course Description

Based on "Fundamentals of Complex Analysis, with Applications to Engineering and Science", by E.B. Saff and A.D. Snider (3rd Edition). "Visual Complex Analysis", by Tristan Needham, is also referred to a lot. Mathematica is often used, especially to visualize complex analytic (conformal) mappings.

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