# 07/22/2020: Complex Analysis (Theodore W. Gamelin) /home/zack/Dropbox/Library/Theodore W. Gamelin/Complex Analysis (802)/Complex Analysis - Theodore W. Gamelin.pdf Last Annotation: 07/22/2020 ## Highlights - conformal self-maps of the open unit disk have the form (Theodore W. Gamelin 307) - Sectors\. A sector can be mapped onto a half-plane with the aid of the power function z°, (Theodore W. Gamelin 308) - D = {0 < argz < a}, (Theodore W. Gamelin 308) - The fractional linear transformation w = \(2 —4\)/\(z + i\) maps the open upper half-plane H onto the open unit disk \. (Theodore W. Gamelin 308) - its inverse z = ¢\(1 + w\)/\(1 — w\)\. (Theodore W. Gamelin 308) - 2T/ g w—‘ﬂ\(z\)—m, z € (Theodore W. Gamelin 308) - The exponential function e¢\* maps horizontal strips to sectors\. (Theodore W. Gamelin 309) - Since fractional linear transformations map circles to circles, the images of the two arcs lie on circles passing through 0 and oo, and so are rays from 0 to co\. (Theodore W. Gamelin 310) - fractional linear transformation { = g\(z\) mapping zo to 0 and 23 to co\. (Theodore W. Gamelin 310)