MATH253 Chapter Notes - Chapter 1-24: Arnold Tongue, Bilinear Map, Joule

• School: University of Alberta
• Department: Mathematics
• Course: MATH253
• Professor: Souvik Goswami

Document Summary

Math 311 lec a1 - spring 2015; fill in for the class lectures, from james lewis"s informal lecture notes. Special case i: fix a, b c, a (cid:54)= 0, and consider the map w = az + b. This is a special case of maps of the form: az + b cz + d ad bc (cid:54)= 0, called linear fractional transformations (lft"s). have c = 0 and d = 1. Thus a (cid:54)= 0 ad bc (cid:54)= 0. ] [in the case w = az + b, we. Proposition 0. 1. w = az + b maps lines to lines and circles to circles. Q w b a z = (cid:12)(cid:12)(cid:12)(cid:12) w b a. P i. e. the corresponding line w((cid:96)) in the w-plane is given by the locus |w w(p)| = Circles: suppose c given by |z p| = r. then if z is on c: (cid:12)(cid:12)(cid:12)(cid:12) w b a (cid:12)(cid:12)(cid:12)(cid:12) = r, |w (ap + b) (cid:124) (cid:123)(cid:122) (cid:125) w(p)