MATH253 Chapter Notes - Chapter 1-24: Product Rule, Positive Real Numbers, Analytic Function

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

Document Summary

From james lewis"s informal lecture notes, with permission from. Math 311 lec a1 - spring 2015; the author. Conformal maps w = f (z) is said to be conformal if it preserves angles. Suppose w = f (z) is analytic1 at p c and that f(cid:48)(p) (cid:54)= 0. Let zj(t) (j = 1, 2, t r) be two paths passing through p, more explicitly, suppose j(0) (cid:54)= 0. Consider the angle created by the two vectors zj(0) = p and that z(cid:48) z(cid:48) Then f (zj(t)) (j = 1, 2) are two paths passing through f (p). I (0) is claim that the angle created by the vectors(cid:0)f (z1(t))(cid:1)(cid:48) (0) and(cid:0)f (z2(t))(cid:1)(cid:48) also = . To see this observe by the chain rule that. Thus arg(cid:0)(cid:0)f (z2(t))(cid:1)(cid:48) (cid:0)f (zj(t))(cid:1)(cid:48) (0)(cid:1) arg(cid:0)(cid:0)f (z1(t))(cid:1)(cid:48) = arg(z(cid:48) (0) = f(cid:48)(zj(0))z(cid:48) j(0) = f(cid:48)(p)z(cid:48) j(0). (0)(cid:1) = arg(cid:0)f(cid:48)(p)z(cid:48) In short, f acts on directions at p by rotation by arg(f(cid:48)(p))