MATH 113 MIDTERM EXAM: SOLUTIONS (Complete And Verified Study material) (7pages) LEARNEXAMS

(1) Consider the map f(z) = 1/z. (a) For which points z ∈ C does f shrink a sufficiently small neighborhood of z? That is, for which points z is limǫ→0 Area(f(Dǫ(z))) Area(Dǫ(z)) < 1 xss=removed> 1. (b) If z = x + iy, what are the images of the lines x = const and the lines y = const? Draw the images and identify the geometric objects they trace out, with proof. The point z = a + iy on the line x = a is mapped to the point 1 z = a − iy a 2 + y 2 , with real and imaginary parts u = a a 2 + y 2 , v = − y a 2 + y 2 . These satisfy u 2 + v 2 = a 2 + y 2 (a 2 + y 2) 2 = 1 a 2 + y 2 = u a or, equivalently, u − 1 2a 2 + v 2 = 1 4a 2 which is the equation of a circle with radius 1/2|a| and center (1/2a, 0); that is, the image of a vertical line is a circle with center on the x-axis, passing throught the origin. Similarly, the image of the horizontal line y = b is a circle of radius 1/2|b| with center at (0, −1/2b). (c) What is the image of the first quadrant? Where does the boundary of the first quadrant go (with orientation)?