Continuing with the examples 2 g fz z is conformal

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Unformatted text preview: oshy + i cosx sinhy and we find out that it does this Æ –π/2 π/2 –1 1 the next block over (π/2 ≤ x ≤ π) goes underneath the axis, and then it repeats as we go across the left-hand 1 1 or w = . z z Look at what happens to the general point z = x + iy 1 x - iy w= = = u + iv x + iy x2 + y2 A vertical line in the w-plane corresponds to u = k x 2 2 = k, a constant x +y (G) f: C -{0}ÆC ; f(z) = I I 1 But this is the equation to a circle For instance, taking k = 2 gives the circle center (1, 0) radius 1. In general, all these circles pass through the origin (where f is not defined)., since the above equation, when cross-multiplied, is satisfied by (0, 0). Similarly, horizontal lines also correspond to circles, but this time centered on the y-axis. In general, we have the following: 21 Proposition 6.4 The transformation w = 1/z takes circles or straight lines to circles or straight lines. Proof One can represent circles and straight lines by 22 A(x +y ) + Bx + Cy + D = 0 2 2 Now x + y =...
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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

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